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First Memoir 
On some Properties of the 
Perfect Positive 
Quadratic Forms 


Georges Fedosevich Voronoi 


Monograph Translations Series 
Series Editor: Vaen Sryayudhya 


Translated by 
Kit Tyabandha, Ph.D. 

God’s Ayudhya’s Defence 
Bangkok 


7 th February, 2007 



Catalogue in Publication Data 
G. F. Voronoi 

On some Properties of the Perfect Positive Quadratic Formes: - Bangkok, Kittix, GAD, 
2007 

116 p. 

1. On some Properties of the Perfect Positive Quadratic Formes. I. Voronoi, G. F. II. 
Tyabandha, Kit III. Mathematics. 

510 

ISBN 978-974-88248-9-5 


Translation © Kit Tyabandha, 2007 
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G. F. Voronoi (1908) 


Monograph Translations Series 


Preface 


George Fedosevich Voronoi was born on 28 th April 
1868 , in Zhuravka, Poltava Guberniya, Russia, which 
is now in Ukraine. He died on 20 th November 1908 in 
Warsaw, Poland. Both his master’s degree, 1894, on the 
algebraic integers associated with the roots of an irre¬ 
ducible cubic equation and his doctoral thesis on algo¬ 
rithms for continued fractions were awarded the Bun- 
yakovsky prize by the St. Petersburg Academy of Sci¬ 
ences. But he decided that he wanted to teach at the 
Warsaw University where he extended work by Zolotarev 
on algebraic numbers and the geometry of numbers. He 
met Minkowski in 1904 at an international conference in 
Heidelberg. 

The three papers by Voronoi all appeared in the same 
journal, the influential and prestigious Journal filr die 
reine und angewandte Mathematik [Journal for the pure 
and applied mathematics]. This journal was a leading 
mathematical journal during 19 th and early 20 th cen¬ 
turies when most of its publications appeared either in 
French or in German. In French it is called Journal de 
Crelle or Crelle’s Journal after the name of its founder 
in 1826 by August Leopold Crelle (1780—1855). Orig¬ 
inally Crelle intended the journal to emphasise equally 
both pure and applied mathematics. But the policy soon 
changed and it has been dealing solely with pure mathe¬ 
matics since the start of his second and short-lived Jour¬ 
nal fur die Bankunst (1829—1851) to deal the application 
side. 

The main idea is quite simple. It is that space can 
be partitioned into a set of regions, each surrounding a 
single point which is sometimes called a nucleus. Every 
point in a given region is closer to its own nucleus than 
to any nuclei of other regions. This idea has found so 
many applications in nature that I think it is as beau¬ 
tiful as the golden ratio 1+ f^ is. It has been used to 
study the forest fire and it has been used to study the 
structure of the distribution of galaxy. In fact the inter- 

God’s Ayudhya’s Defence 7 th February, 2007 i 



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G. F. Voronoi (1908) 


nal structure of many things has proved to be Voronoi, 
things in nature as well as man-made ones, for example 
plant cells and filtering membranes. Looking at a two- 
dimensional Voronoi structure will remind one of plant 
or animal cells partitioned by straight walls which re¬ 
sult from cosiness of circular cells growing among their 
neighbours. 

Voronoi’s lifelong work was in theory of numbers and 
is divided into three groups, namely algebraic theory of 
numbers, analytic theory of numbers, and geometry of 
numbers. His three papers translated here make up two 
of the planned series of memoirs to apply the principal 
of Continuous Hermite (Charles Hermite, 1822—1901) pa¬ 
rameters to problems of the arithmetical theory of def¬ 
inite and indefinite quadratic forms. He had completed 
only two of the series when he died in Warsaw, Poland 
on 20 th November 1908 , before the last one appeared in 
print in 1909. A short obituary written by Kurt Hensel 
(1861 — 1941) was included at the end of the paper, which 
is also included here in translation. 

The first paper gives characteristics of complete qua¬ 
dratic forms. In it Voronoi solved the question posed by 
Hermite on the upper limit of the minima of the posi¬ 
tive quadratic forms for a given discriminant of n vari¬ 
ables. Zolotarev (Egor Ivanovich Zolotarev, 1847—1878) 
and Korkin (Aleksandr Nikolaevich Korkin) had given 
solutions for n = 4 and n = 5. Voronoi gave an algo¬ 
rithmic solution for any n. He did this with the help 
of the methods of the geometrical theory of numbers. 
The present volume gives an English translation of this 
paper. 

The second and the third papers deal with simple 
parallelepipeds, that is polyhedra with parallelograms as 
all their faces. He gave the determination of all possi¬ 
ble methods of filling an n-dimensional Euclidean space 
with identical convex non-intersecting polyhedra (par¬ 
allelepipeds) which have completely contiguous bound¬ 
aries. A solution of this problem for three-dimensional 
space had been given by Fedorov (Evgraf Stepanovich 
Fyodorov, 1853—1919) who was a crystallographer al¬ 
though the proof he gave is said to be incomplete. Min- 


li 


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kowski (Hermann Minkowski, 1864—1909) showed in 1896 
that the parallelepipeds must have centres of symmetry. 
He also demonstrated that the number of their bound¬ 
aries did not exceed 2(2 n — 1). Voronoi imposed further 
the requirement that n + 1 parallelepipeds converge at 
each summit and solved the problem for these conditions 
completely. 

Voronoi’s collected works appeared in three volumes 
under the title of Sobranie sochineny, Kiev, 1952—1953. 
He is sometimes referred to as belonging to the St. Pe¬ 
tersburg school of the Theory of Numbers. This must 
have been the Petersburg Mathematical School some¬ 
times called Chebyshev School or Petersburg School. It 
was founded by Chebyshev (Pafnuty Lvovich Chebyshev, 
1821—1894) and had prominent figures as Grave (Dmitri 
Aleksandrovich Grave), Krylov (Aleksei Nikolaevich Kr¬ 
ylov, 1863—1945), Lyapunov (Aleksandr Mikhailovich L- 
yapunov, 1857—1918), Markov (Andrei Andreyevich Mar¬ 
kov, 1856—1922), Sohotski (Yulian-Karl Vasilievich So- 
khotsky, 1842—1927), Steklov (Vladimir Andreevich Ste- 
klov, 1864—1926), Korkin, K. A. Posse, and A. V. Vas- 
siliev. 


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G. F. Voronoi (1908) 


iv 


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G. F. Voronoi (1908) 


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New applications of continuous parameters 

to the 

theory of the quadratic form 

First Memoir 
On some properties of the 
perfect positive quadratic forms 

by 

Mr. Georges Voronoi in Warsaw 

[Journal fur die reine und angewandte Mathematik] 

[V. 133, p. 97-178, 1908] 

[translated by K N Tiyapan] 
Introduction 


Hermite had introduced in the theory of numbers a 
new and fruitful principle, namely: being given a set ( x ) 
of systems (x\ , X 2 , ■ ■ ■ , x n ) for all the values of xi,X 2 ,---> 
x n , one associates with the set ( x ) a set ( R ) composed of 
the domains in a manner such that by studying the set 
(i?) one studies at the same time the set ( x ) . 

Hermite has shown f numerous applications of the 

f Hermite. Extraits de lettres de M. Ch. Hermite a M. 
Jacobi sur differents objets de la theorie des nombres. 
[Excerpts from letters of Mr. Ch. Hermite to Mr. Ja¬ 
cobi on various subjects in the theory of numbers] (This 
Journal V. 40, p. 261) 

Hermite. Sur l’lntroduction des variables continues dans 
la theorie des nombres. [On the introduction of the con¬ 
tinuous variables in the theory of numbers] (This Journal 
V. 41, p. 191) 

Hermite. Sur la theorie des formes quadratiques. [On 
the theory of quadratic forms] (This Journal V. 47, p 
313) 


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new principle to the generalisation of continuous frac¬ 
tions, to the study of algebraic units, etc. 

The ideas of Hermite have been developed in the 
works of Mr.’s Zolotareff, Charve, Selling, Minkowski. 

X 


I intend to publish a series of Memoires in which I 
shall show new applications of the principle of Hermite 
to the various problems of the arithmetic theory of def¬ 
inite and indefinite quadratic forms. 

In this Memoire, I study the properties of the min¬ 
imum of positive quadratic forms and of their various 
representations by systems of integers. 

Hermite has discovered an important property of the 
minimum M of positive quadratic forms J2 a ij x i x j i n n 
variables and of the determinant D, namely: 

n-1 

M<(f) - VD, 

and he has demonstrated numerous applications of this 

$ Zolotareff. On an indeterminate equation of the third 
degree (Petersbourg, 1869, in Russian.) 

Zolotareff. Theory of complex integers with applications 
to the integral calculus. (Petersbourg, 1874, in Russian.) 

Charve. De la reduction des formes quadratiques ter- 
naires positives et de leur application aux irrationelles 
de troisieme degre. [Of the reduction of positive ternary 
quadratic forms and of their application to the irration¬ 
als of third degree] (Suppl. to V. IX of Annales Scien- 
tifiques de l’Ecole Normale Superieure, 1880) 

Selling. Uber die binaren und ternaren quadratischen 
Formen. [On the binary and ternary quadratic forms] 
(This Journal, V. 77, p. 143) 

Minkowski. Geometrie der Zahlen. [Geometry of num¬ 
bers] (Leipzig, 1896) 


6 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 

formula. 


Monograph Translations Series 


In a letter to Jacobi, Hermite has said §: 

“That which precedes sufficiently indicates an infin¬ 
ity of other analogous consequences which, all, will de¬ 
pend on the difficult study of an exact limit of the min¬ 
imum of any definite form. Thereupon I then form only 
one conjecture. My first studies in the case of a form in 
n variables of the determinant D have given me the limit 

re — 1 

(f)^~ Vd, I am inclined to presume, but without being 

re -1 

able to demonstrate that the numerical coefficient (|) 2 
has to be replaced by y== ” 

Mr.’s Korkine and Zolotareff has under taken the 
study of the exact limit of the minimum of positive quad¬ 
ratic forms of the same determinant. 


By indicating with M(aij ) the minimum and with 
D ( dij ) the determinant of the form Y&ijXiXj , one will 
have the minimum 


M(aij ) 


M (cLjj ) 
y/D(dij ) 


of a positive quadratic form with determinant 1. 


By virtue of the theorem of Hermite the function 
M(aij) verifies the inequality 

< (|)^« 

§ This Journal. V 40, p. 296 

f Mr. Minkowski has demonstrated an upper limit of 
the function M(aij) 

M(aij) < n 

much simpler than that from Hermite. 

( Minkowski. Uber die positiven quadratischen For- 
men und uber kettenbruchahnliche Algorithmen. [On 
the positive quadratic forms and on continued fraction 
algorithm] This Journal V. 107, p. 291) 

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therefore it is bounded within the set (/) of all the pos¬ 
itive quadratic forms of real coefficients. 

Mr.’s Korkine and Zolotareff have demonstrated f 
that the function _M(aqj) possesses many maxima in the 
set (/) which correspond to the various classes of equiv¬ 
alent positive quadratic forms. 

The limit y^ l+1 indicated by Hermite in the letter to 
Jacobi (source cited) is only a maximum value of the 
function M(aij). 

The binary and ternary positive quadratic forms pos¬ 
sess a single maximum which is therefore, in this case, 
the exact limit of values of the function M(aij). 

Reckoning from the number of variables n > 4, one 
meets many maxima of the function Af(ajj). 

Mr.’s Korkine and Zolotareff have found many values 
of various maxima of the function M(oqj) which exceed 
the limit 2 . „ indicated by Hermite, but do not exceed 
the limit 2. 

The study of the exact limit of the minimum of pos¬ 
itive quadratic forms of the equal determinant comes 
down, after Mr.’s Korkine and Zolotareff, to the study 
of all the various classes of positive quadratic forms to 
which correspond the maximum values of the function 
M(aij). 


The maximum maximorum of values of the function 
M(aij ) is the largest value of the function M(aij) which 
presents a numerical function as p,(n). 

Mr.’s Korkine and Zolotareff have determined the 
following values of the function p,(n): 

M2) = y|, M3) = #2, M4) = M5) = VE, • 

f Korkine and Zolotareff. Sur les formes quadratiques. 
[On the quadratic forms] Mathematische Annalen, V. VI, 
p. 366 and V. XI, p. 242 


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G. F. Voronoi (1908) 


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They have called extreme the quadratic forms which 
yield to the function Af(a^) a maximum value. 

The extreme quadratic forms enjoy an important pr¬ 
operty, namely: 

I. Any extreme quadratic form is determined by the 
value of its minimum and by all the representations of 
the minimum. 

Mr.’s Korkine and Zolotareff have determined all the 
classes of extreme forms in 2, 3, 4 and 5 vertices. 

By studying these extreme forms, I have observed 
that they are all well defined by the property (I). There 
is only reckoning from positive forms in six variable wh¬ 
ich I have encountered positive quadratic forms which 
enjoyed the property (I) and are not of extreme forms. 

I call “perfect” any positive quadratic form which 
enjoys the property (I). 

I demonstrate that the set of all the perfect forms 
in n variables can be divided into classes the number of 
which is finite. 

All extreme form being, by virtue of the property 
I, a perfect form, it results in that the function ju(n) 
presents the maximum of values of the function M(aij) 
which correspond to the various classes of perfect forms. 

I have established an algorithm for the search of var¬ 
ious perfect forms by introducing a definition of contigu¬ 
ous perfect forms. 

To that effect, I make correspond to the set (<p) of 
all the perfect forms in n variables a set (f?) of domains 
in n bH~ 1 ) dimensions determined with the help of linear 
inequalities. 

The set (f?) of domains in n ( n + 1 '> dimensions presents 
a partition of the set (/) of all the positive quadratic 


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forms in n variables. 

Each domain R possesses in the set ( R ) a contiguous 
domain which is well determined by any one face in - 
n ( n + 1 ') — i dimensions of the domain R. 

I demonstrate that the domain R corresponding to 
the perfect form tp(x±, x 2 , ■ ■ ■ , x n ) being determined by the 
linear inequalities 

^Pifaij >0, (fe = 1,2, . . ., ct) 

one will have a perfect forms defined by the equalities 

<Pk(xi, X2, ■ ■ ■ , X n ) = (p(x!, X 2 , ■ ■ ■ , X n ) + Pk^k(xi, X 2 , ■ ■ ■ , X n ), 

(k= 1,2,...,a) (1) 

where 

^k(xi,x 2 , ■ ■ ■ ,x n ) = 'E'P^XiXj, 

provided that the positive parameter p *. (k = 1,2, . . ., a) 
presents the smallest value of the function 

<t>(xi,x 2 , . . . , x n ) - M 
-<P(x 1 ,x 2 , • • • , x n ) 

where (xi, x 2 , . . . , x n ) < 0 and M is minimum of the form 
4>(xi,x 2 , . . . , x n ). 

I call “contiguous to the perfect form <fi(xi, x 2 , . . . , 
x n )” the perfect forms (1). 

Any substitution in integer coefficients and with de¬ 
terminant ±1 belonging to the group g of substitutions 
which do not change the form cf> permute only the forms 
(1). One can, therefore, divide the forms (1) into classes 
of equivalent forms with the help of substitutions of the 
group g. By choosing one form in each class, one will 
have a system of perfect forms contiguous to the perfect 
form (p which can replace the system (1). 

By proceeding in this manner, one can obtain a sys¬ 
tem complete of representatives of various classes of per¬ 
fect forms. 

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G. F. Voronoi (1908) 


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The corresponding domains will form complete sys¬ 
tem of representatives of various classes of the set ( R ). 

I have remarked that a similar system 

R, Ri, f?2, • • • J fir-1 (2) 

of domains of the set (f?) can serve towards the reduction 
of positive quadratic forms. 

I call reduced any positive quadratic form belonging 
to one of the domains (2). 

It results from this definition: 

I. Any positive quadratic form can be transformed into 
an equivalent reduced form, with the help of a substitution 
which presents a product of substitutions belonging to a 
series of substitutions 


Si , S 2 , , S m 

which depend only on the choice of the system (2). 

II. Two reduced forms can be equivalent only provided 
that the corresponding substitution belonged to a series of 
substitutions the number of which is finite. 

The weak point of the new method of reduction of 
positive quadratic forms, demonstrated in this Memoire, 
consists in that the number of substitutions which trans¬ 
form into itself the domains of the set (f?) or their faces 
is, in general, very large. 

The application of the general theory demonstrated 
in this Memoire to the numerical examples will be partic¬ 
ularly facilitated if one knew how to solve the following 
problem: 

Being given a group G of substitutions which trans¬ 
form into itself a domain R, one would like to partition 
this domain into equivalent parts the number of which will 
be equal to the number of substitutions of the group G and 


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on condition that the number of faces in n ( n +D — 1 dimen¬ 
sions of domains obtained be the smallest possible. 

I show in this Memoire the solution of the problem 
introduced in two cases: n = 2 and n = 3. 

From the number of variable n > 4, I do not know 
any practical solution of the problem posed. 

First Part 
General theory 
of 

perfect positive quadratic forms 
and 

domains which correspond to them 

Definition of perfect quadratic forms. 


Let 

<t>(xi, x 2 , ■ ■ ■ , x n ) = y ^ajjXjXj (1) 

be any positive quadratic form. By indicating with 

Oil) 1 21) • • • ) Ini') i (l 12, 122, • • • On 2 ), ■ ■ ■ , (l Is, hs, ■ ■ ■ , Ins) (2) 

the various representations of the minimum M of the 
form J2aijXiXj, one will have the equalities 

^ aijUkljk = M, (k = 1,2, . . ., s) (3) 


One will not consider in the following the two sys¬ 
tems 


(Ilk, l2k, • • • ) Ink) and ( Ilk, l2k, • • • , Ink), 

(k= 1,2 

as different and one will arbitrarily choose one of these 
systems. 

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On the ground of the supposition made, one will have 
the inequality 

^2 CLijXiXj > M 

provided that a system (xi, X 2 , ■ ■ ■, x n ) of integer values 
of variables x\, x?, . . . , x n did not belong to the series (2), 
excluding the system x\ = 0, X 2 = 0, . . . , x n = 0. 

By considering the equalities (3) as the equations 
which serve to determine n ^ n 2 +1 ' 1 coefficients of the quad¬ 
ratic form 'ZaijXiXj, one will have only two cases to ex¬ 
amine: 


1. ) there exist a finite number of solutions of equa¬ 
tions (3), 

2. ) the equations (3) admit only a single system of 
solutions. 


Let us examine the first case, 

Let us suppose that there exists an infinite number 
of solutions of equations (3). 

One will find in this case an infinite number of values 
of parameters 

Pij = Pjii ( ^ = 1 j 2, . . . , n ; j = 1,2,...,77.) 

verifying the equations 

^ ' Pijlikljk ^ 0, (fc = 1, 2, . . . , s) (^) 

the values p^j = 0, i = 1,2, . . ., n; j = 1,2, . . ., n being ex¬ 
cluded. 

By indicating 

'&(x 1 ,X 2 , ■ ■ ■ ,X n ) = ^PijXiXj , 

let us consider the set of positive quadratic forms deter¬ 
mined by the equality 

f(x 1 ,X 2 ,---,X n ) = p(x 1 ,X 2 ,---,Xn) + P^(x 1 ,X 2 ,---,X n ), (5) 

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the parameter p being arbitrary. 

For a quadratic form determined by the equality (5) 
to be positive, it is necessary and sufficient that the cor¬ 
responding value of the parameter p be continuous in a 
certain interval 

-R! < p < R. 


It can turn out that R = +oo, in this case the lower 
limit —R 1 will be finite. By replacing in the equality (5) 
the form A/ (x±, x 2 , . . . , x n ) by the form — SE^xi, X 2 , . . . , x n ), 
that which is permitted by virtue of (4), one will have 
the interval 

-R < p < R 1 , 

therefore one can suppose that the upper limit R is fi¬ 
nite. 


The corresponding quadratic form, determined by 
the equality 

f(x!,x 2 , . . . ,x n ) = (p(x 1} X 2 , . . . ,x n ) + RA?(x 1,052, . . . , X n ) , 


will not be positive, but it will not have negative values 
either; one concludes that least for a system (£i,£ 2 , . .., 
£ n ) of real values of variables x\, x 2 , ■ ■ . , x n the form /(x i, 
x 2 , . . • ,x n ) attains in its value the smallest which is zero, 
and it follows that the system (£i, £ 2 , ■ ■ ■ ; £ n ) verifies the 
equation 


df_ 

d£,i 




= 0. 


(i = 1,2, . . ., n) 


By eliminating from these equations £ 1 , £ 2 , ■ ■ ■, €n one 
obtains the equation 



an + -R-Pn, 

ai 2 + RP 12 , ■ ■ 

• j ^ln “h RP±n 


D(R ) = 

a 2 i + RP 21 , 

a> 22 + RP 22 , ■ ■ 

• j a 2n + RP2n 

= 0 


a n i “l - RRn i> 

®n2 “1“ RPn2i • • 

• j Q'nn “1“ RPnn 



The smallest positive root of this equation presents 
the value of R searched for. 


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G. F. Voronoi (1908) 


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Let us examine the set (/) of positive quadratic forms 
determined by the equality (5) with condition 


0 < p < R. 


( 6 ) 


Theorem. To the set (/) belongs a quadratic form pi(x±, 
X 2 , ■ ■ ■ ,x n ) which is well determined by the following con¬ 
ditions: 


1. all the representations of the minimum M of the 
form p(xi, X 2 , ■ ■ ■, x n ) are also representations of the min¬ 
imum M of the form <pi(xi, x ^, • • • , x n ), 

2. the form cpi(xi, X 2 , ■ ■ ■, x n ) moreover possesses at 
least another representation of the minimum M. 

Let us indicate by M(p) the minimum and by D(p) 
the determinant of the quadratic form f(x 1 , 2 : 2 , 
defined by the equality (5) with condition (6). 

By virtue fo the theorem by Hermite, one will have 
the inequality 

M(p)<p(n)tfD(pj. (7) 


We have demonstrated that D(R ) = 0, it results in 
that a value of the parameter p can be chosen in the 
interval (6) such that the inequality 

p(n) V-D(p) < M 

holds. One will have, because of (7), 

M (p) < M. (8) 


Let us indicate by (li, I 2 , ■ ■ ■ , l n ) a representation of 
the minimum M(p) of the form f (x 1 , X 2 , ■ ■ ■ , x n ) verifying 
the inequality (8). 

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One will have 

I2, ■ ■ ■ , In) + P'S' Ol, I2, • • • , In) < M, ( 9 ) 

and as a result 

<p(h,h, ■ ■ ■ ,l n ) > M and d>(Zi, l 2 , ■ ■ ■, l n ) < 0 . (10) 

This posed, let us find the smallest value of the func¬ 
tion 

<p(xi, x 2 , ■ ■ ■, x n ) - M 
-^/(x 1 ,x 2 , • • • , X n ) 
determined with condition 

^(zi, x 2 , ■ ■ ■ , x n ) < 0. 


( 11 ) 

( 12 ) 


To that effect, let us examine the inequality 
V(xi,x 2 , • ■ ■ , x n ) - M < (p(h,l 2 , ■ ■ ■ , l n ) ~ M 

^ (x 1, X 2 , ... , X n ) *(h,l 2 ,...,ln) 


By virtue of (9), (10) and (12), one will have 

<p(x 1} x 2 , . . . ,x n ) + p'&(x 1 ,x 2 , . . . ,x n ) < M. 


The quadratic form p(x\,x 2 , ■ ■ ■ , x n ) + pd/( x±,x 2 , . . . , 
x n ) being positive, there exists only a limited number of 
integer values of x±,x 2 ,...,x n verifying this inequality. 
Among these systems are found all the systems which 
give back to the function (11) the smallest value deter¬ 
mined with condition (12). 


Let us indicate by 


(it it 11 \ (ltt 111 lit \ (](r) j(r ) /(O) 

ri) ) 4 n / ’ \ l l’ • • • ) ’ ’ • • ) r 1 ?‘ , 2 > • • • ? 1 n ) 


all the representations of the positive minimum pi of the 
function (11). 


By declaring 

<Pl(xi, X 2 , . . . ,X n ) = p(xi,x 2 , . . . ,X n ) + pi'fy (xi, x 2 , . . . ,x n ), 
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G. F. Voronoi (1908) 


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one obtains the positive quadratic form ip±(xi, x 2 , • • •, x n ) 
the minimum M of which is represented by the systems 
(2) and (13), this is that which one will demonstrate 
without trouble. 


With the help of the procedure previously shown, one 
will determine a series of positive quadratic forms 


<P,<Pi,<P2, ■ ■ ■ (14) 

which enjoy the following property: by indicating with 
Sk the number of representations of the minimum of the 
form <Pk(k = 1,2,...), one will have the inequalities 


s < Si < S 2 < ■ ■ ■ 


(15) 


A similar series of positive quadratic forms of n vari¬ 
ables can not be extended indefinitely, this is that which 
we will demonstrate with the help of the following lem¬ 
ma. 


Lemma. The number of various representations of the 
minimum of a positive quadratic form in n variables does 
not exceed 2 n — 1. 


Let us indicate by (Zi, l 2 , ■ • •, l n ) and (l ), 1' 2 , . ■ . , l ' n ) any 
two representations of the minimum M of the positive 
quadratic form Y) a ij x i x j ■ 

Let us suppose that by declaring 

l' i = l i + 2t i , (i = 1,2, . . ,,n) (16) 

the number ti,t 2 ,...,t n would be integer. 

As 


^ aijljl'j = M and aijljlj = M, 

by virtue of (16), it becomes 

ijlitj + E Gjijt it j — 0 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 17 



Monograph Translations Series 


G. F. Voronoi (1908) 


One will present this equality under the form 
FI Q ij ~l ~ tj) (lj tj') aijtitj = aijlilj. (17) 


By noticing that 

y y Q’jji'ii'j ^ y ^ etjjiji j j 

one finds, by virtue of (17), 

F aij(li + tj)(lj + tj ) < 0, 

therefore it is necessary that 

y ' tj)) (lj tj) — 0, 

and consequently 

li + tj = 0. (i = 1, 2, . . . , n) 

Because of (16), one obtains 

l j = — li ■ (i = 1, 2n) 

This posed, let us divide the set (X) of all the systems 
(xi, X 2 , . . . , x n ) of integer values of xi,X 2 ,...,x n into 2 n 
classes, with regard to the modulo 2. 

We have demonstrated that two different representa¬ 
tions of the minimum M of the form ^djjXjXj will not 
belong to the same class; neither will any representa¬ 
tion of the minimum M belong to the class made up of 
systems (x±, X 2 , ■ ■ ■, x n ) satisfying the condition 

a3j=0(mod2), (i = 1, 2, . . . , n) 

therefore the number of various representations of the 
minimum of a positive quadratic form can not be greater 
than 2 n - 1. 


We have demonstrated that the series (14) of positive 
quadratic forms satisfying the condition (15) can not be 
extended indefinitely, therefore the series (14) will be 
terminated by a form ip/- which enjoys the following prop¬ 
erty: the form <is determined by the representations 
of its minimum. 


18 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


Definition. One will call perfect any positive quadratic 
form which is determined by the representations of its 
minimum. 


Let us suppose that the form (1) be perfect, one will 
have in this case only a single system of solutions of 
equations (3). 

On the ground of the supposition made, the equa¬ 
tions _ 

^ ' Pijlikljk — Oj (h — 1,2, . . . , s) 
admit only a single system of solutions 

Pij = Pji = 0. (i = 1,2,. . ., n; j = 1,2, . . ., n) 

By effecting the solution of equations (3), one ob¬ 
tains the equalities 

aij — otij AT , (i — 1,2, . . ■ , n ■ J — 1,2, . . ., n) 

where the coefficients ctij are rational. 


It results in that the perfect form is of rational 
coefficients. In the following one will not consider as 
different the perfect forms of proportional coefficients. 

Fundamental properties of perfect quadratic forms. 


Let 

P ( x i, x 2, . . . , n) — aijXiXj 

be a perfect quadratic form. Let us suppose that all the 
different representations of the minimum of the perfect 
form cp make up the series 

(^11)^21) • • • j lnl) i (l12i 122i ■ ■ ■ i ln2 ),...,(hs,l2s,...,l n s)- (1) 


By choosing any n systems in this series, let us ex¬ 
amine the determinant 


' 

'Ilf 

ll 2, 

• • • , l In 



> 

'21, 

122, 

• • • , ^2n 

= ±LU. 

( 2 ) 

n 1, 

l n2 j 

• • • , Inn 




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G. F. Voronoi (1908) 


All the determinants that one can form this way can 
not cancel each other out. By supposing the contrary, 
one will have s equations of the form 

n — 1 

lik = Y, li r uj k) , (i = 1,2, ... ,n-,k = 1,2, ... ,s) (3) 

r= 1 


One will choose a system of n( ' n + 1 ' > parameters pij = 
Pji verifying n ( n + 1 '> equations 

^ ' Pijlirljt — Oj (T — 1,2, . . . , 71 1 ] t — 1, 2, . . ., n 1) 

and by virtue of (3), one will have 

^ ' Pijlikljk — dj (k — 1,2, • • • , £») 
which is impossible. 

The numerical value uj of the determinant (2) can not 
exceed a fixed limit. To demonstrate this, let us effect 
a transformation of the perfect form ip with the help of 
a substitution 


Xi — l ir*E r , (1 — 1, 2 , . . . , 71) 

r= 1 

one will obtain a form 

¥>'(z'i, z' 2 , • • • , x‘ n ) = Y^a’ijx’ix’j, 

where 

a’a = M. (i = 1,2, ..., n ) 


(4) 


(5) 


By indicating with D' the determinant of the form 
p>', one will have the inequality 

a ll a 22 ’ ’ ’ a nn — D , 

by virtue of the known property of positive quadratic 
forms. 

Considering (5), one obtains 

M n > D'. (6) 

20 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


By indicating with D the determinant of the form ip, 
one will have, because of (2) and (4), 

D' = Du 2 , 

therefore the inequality (6) reduces to the one here: 

Du 2 < M n . 


By virtue of the theorem by Hermite, one has the 
inequality 

M < /u(n) \fl5 ; 


it follows that 

(?) 


Any perfect form will obviously be transformed into 
a form, also perfect, with the help of all linear substitu¬ 
tion of integer coefficients and of determinant ±1. 

One concludes this that there exists a finite a finite 
number of equivalent perfect forms. 

The set (p) of all the perfect forms in n variables can 
be divided into different classes provided that each class 
be made up of all the equivalent perfect forms. 


Theorem. The number of different classes of per¬ 
fect forms in n variables is finite. 


Let us indicate by 


A k — 11 k 1 1 2 k & 2 “1“ • • • “1” l nk'E n ( & — 1)2, . . . , S ) 


s linear forms 

Ai, A 2 , . . . , A s (8) 

f See the Memoire of Mr.’s Korkine and Zolotareff sur 
les formes quadratiques positives. (Mathematische An- 
nalen V. XI, p. 256) 


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Monograph Translations Series G. F. Voronoi (1908) 

which correspond to the systems (1) of representations 
of the minimum of the form SI/. 

One establishes this way a uniform correspondence 
between a perfect form p and the system (8) of linear 
forms. 

Let us suppose that one had transformed the per¬ 
fected form p with the help of a substitution S by inte¬ 
ger coefficients and with determinant ±1, one will obtain 
an equivalent perfect form p'. Let us indicate by 

Ai, A' 2j ...,A' s (9) 

the corresponding system of linear forms. 

One will easily demonstrate that the substitution T, 
adjoint to the substitution S §, will transform the system 
(8) into a system (9). 

One concludes that a certain reduction of perfect 
forms can be effected with the help of the reduction of 
corresponding systems of linear forms. 

The reduction of the system (8) comes down, by virt¬ 
ue of (7), to the reduction of any n linear forms 

Ai, A 2 , • • • , A n (10) 

belonging to the system (8) and with determinant ±u; 
which does not cancel each other out. 

§ The substitution S being defined by the equalities 

n 

Xi = ^2 ctikx'k, (i = 1, 2 , . . ., n) 
k =1 

one calls “substitution adjoint to S” the substitution T 
which is determined by the equalities 

n 

) ' OtikX k — X ^. ( i — 1 , 2 , . . . , 71 ) 

fc =1 


22 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


One will determine with the help of the known meth¬ 
od a substitution T which will transform the linear forms 
(10) of integer coefficients into linear forms, 

A' 1 ,A' 2j ... J A' n (11) 

satisfying to the following conditions 

Afc = Pk,kX d - Pk + l,k x k -\-l d - . . . d - Pn,kX n , (fc=l,2,...,Tl) 

< PnP 22 ---Pnn = w and p kk >0, (k = 1,2,... ,n) 

, 0 < p k+i , k < p kk . (i = 1,2, .. . ,n - fc; fc = 1,2,...,n) 


The coefficients of forms (11) being integers, as a 
result they do not exceed fixed limits. 


The substitution T will transform the system (8) into 
a system 

Ai,A' 2 ,...,A' s (12) 

of linear forms. By examining successively the determi¬ 
nants of forms 


(Afc, A 2 , . . . , A'J, (A;, A' fc> ■ ■ ■ , A' n ), . . ., (Ai, A' 2j . . . , A' fc ), 


(k = n-\-l,n-\-2,...,s ) 


one will demonstrate that the numerical values of coef¬ 
ficients of all the linear forms (12) do not exceed fixed 
limits. 

The number of similar systems of linear forms in inte¬ 
ger coefficients being limited, it results in that the num¬ 
ber of different classes of perfect forms is also limited. 


On the domains determined with the help of linear in¬ 
equalities 


We have seen in Number 7 that the study of per¬ 
fect forms can be brought back to the study of certain 
systems of linear forms. 


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G. F. Voronoi (1908) 


One will acquire a new basis to these studies by mak¬ 
ing correspond to each perfect quadratic form in n vari¬ 
ables a domain in n ( n + 1 '> dimensions determined with the 
help of linear inequalities. 

One will address first the general problem by study¬ 
ing the properties of domains determined with the help 
of linear inequalities. J 

Let us consider a system of linear inequalities 
PlkXl + P2kX 2 + • • • + PmkXm > 0, (k = 1,2, . . . , o) 

in any real coefficients. 

One will call point (x) any system (xi, X 2 , • • • , x m ) of 
real values of variables xi, X 2 , ■ ■ ■ , x m and one will indicate 

Vk{x ) = PlkXl + P2kX 2 + • • • + PmkXm■ ( k = 1 , 2 , . . . , (j) 

One will call “domain” the set R of points verifying 
the inequalities 

Uk(x) > 0. (fc= 1,2(1) 

Let us suppose that to the domain R belonged to 
points verifying the inequalities 

Vk(x ) >0, (fc = 1,2, . . . , ct) 

one will call such points interior to the domain R, and 
the domain R will be said to be of m dimensions. 

It can be the case that the domain R does not possess 
interior points. One will demonstrate in this case all 
the points belonging to the domain R verify at least one 
equation 

Vk(x ) = 0, 

the indice h being a value 1,2 , . . ., a. 

It is important to have a criteria with the help of 
which one could recognise whether a domain determined 
by the help of inequalities (1) will be in m dimensions 
or not. 

J See: Minkowski. Geometrie der Zahlen [Geometry of 
the numbers], No. 19, p. 39. 


24 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


Fundamental principle. For a domain determined with 
the help of inequalities (1) to be of m dimensions, it is 
necessary and sufficient that the equation 

^2pkVk(x) = 0 (H) 

k= 1 

did not reduce into an identity so long as the parameters 
p 2 ,p 2 ,...,p a are positive or zero, the values pi = 0 , p 2 = 
0, . . . , pa = 0 being excluded. 


The principle introduced, considered from a certain 
point of view, is evident, but one arrive at the rigorous 
demonstration of this principle only with the help of the 
in depth study of domains determined with the help of 
linear inequalities. 

For more simplicity, one will examine in that which 
follows only domains satisfying the following conditions: 
the equations 


Vk{x) = 0 (k= 1,2, ...,a) (2) 

can not be verified by any point, the point X\ = 0,x 2 = 
0, . . ., x m = 0 being excluded. 

It is easy to demonstrate that the general case will 
always come down to the case examined. 


Definition. One will call edge of the domain R deter¬ 
mined with the help of inequalities (1) the set of points 
belonging to the domain R and verifying the equations 

y k (x) = 0, (k = 1,2, ... ,r where r < a) 

provided that these equations defined the values of xi,x 2 , 
. . . ,x m to an immediate common factor. 


By indicating with (£i,£ 2 , • • • ,£ m ) a point of the edge 
considered, one will determine all the points of the edge 
with the help of equalities 

x 1 — P^i 1 (i — 1 , 2 , . . . , n ) 

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G. F. Voronoi (1908) 


p being an arbitrary positive parameter. 

This results in that each edge of the domain R is well 
determined by any point belonging to it. 

Let us suppose that the domain R possesses s edges 
characterised by the points 

(£fc) = (£lfc>£ 2 fcj'-'j£mfc)' (k = 1,2, . . . , s) 


By declaring 


Xi — ^ ' Pk^ik i (i — 1 , 2 ,..., m ) ( 3 ) 

k = 1 


where 

Pk> 0, (k = 1,2, . . ., s) (4) 

one obtains a point ( x ) belonging to the domain R, the 
positive or zero parameters pi,p 2 ,---,p s being arbitrary. 


Fundamental theorem. Let us suppose that the inequ¬ 
alities (1) which define the domain R satisfy the condition 
(*)■ 


The domain R will be of m dimensions and each point 
belonging to it will be determined by the equalities (3) with 
condition (4) ■ 

The theorem introduced is well known in the case 
m = 2 and m = 3. 

We will demonstrate that by supposing that the the¬ 
orem be true in the case of m — 1 variables, the theorem 
will again be true in the case of m variables. 

Let us examine first the various inequalities of the 
system (1). It can be the case that many among them 
could be put under the form 

S 

Vh(x) = P { k ] yk(x) where p { k h) > 0. 
k= 1 


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G. F. Voronoi (1908) 


Monograph Translations Series 


(k= 1,2,. . .,s-,p[ h) = 0) 


One will call such inequalities dependent and one will 
exclude them from the system (1). 

Let us suppose that the system (1) contained only 
independent inequalities. 

Their number p, on the ground of the supposition (2) 
made, will not be less than m. 

This posed, let us examine a set P k of points belong¬ 
ing to the domain R and verifying an equation 

Vh(x) = 0, (5) 

the indice h having a value 1,2, . . ., a. 

One will call “face of the domain f?” the domain P 

On the ground of the supposition made, the face P k 
will be in m — 1 dimensions. 

To demonstrate this, let us make correspond to any 
point ( x ) verifying the equation (5) a point ( u ) in m — 1 
coordinates («i, U 2 , ■ ■ ■ , by declaring 

m — 1 

Xi = ^2 a ij u j ■ (* = 1,2, . . ., m) 

3 = i 

The system of inequalities (1) will be transformed 
into a system 

Vk(u)> 0 (k = 1,2, . . ., ct; k h) (7) 

of inequalities in m — 1 variables u±, u 2 , . . . , u m _ 1 . 

Let us suppose that one knew how to reduce the 
equation 

<7 

^2 PkVk(u) = 0 where p h = 0 and p k > 0 (k = 1, 2, . . ., a) 
k = 1 

( 8 ) 


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Monograph Translations Series G. F. Voronoi (1908) 

into an identity. By virtue of (6), one will obtain the 
identity 


Y PkVk(x) = py h (x) where p h = 0. 
fc =i 


One can not suppose that p > 0, since otherwise the 
inequality 

Vh(x) > 0 

would be dependent and on the ground of the supposi¬ 
tion made would not belong to the system (1). 

By supposing that p < 0, one will admit ph = — p and 
one will obtain the identity 


<7 

Y PkVk(x) = 0 where p k > 0, ( k = 1, 2, . . . , a) 

k = 1 

which is contrary to the hypothesis. 

We have supposed that the theorem introduced be 
true in the case of m — 1 variables. As the equation (8) 
can not be reduced into an identity, one concludes that 
the system of inequalities (7) defines a domain Bu in m—1 
dimensions. Moreover, by indicating with 

(« 11 , U21 , • • • , U m - 1 , 1 ) , (u 12 , U 22 , 

. . . , Ujn — 1,2)5 ■ ■ ■ j (^ltj • • • 5 — l,t) ( 9 ) 

the points which characterised t edges of the domain B k , 
one will determine any point ( u ) of this domain by the 
equalities 

t 

Ui = Y PkUik where p k > 0, 
k =1 

(k = 1,2,.. ., t; i = 1,2,.. . ,m - 1) (10) 


One will make correspond to the points (9) the points 


((r)=((ln(2r,...,U), (r = 1,2 , . . . , t ) (11) 


28 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


by determining them with the help of equalities (6) and 

(9). 


The points obtained (11) characterise t edges of the 
domain R belonging to the face Ph. Any point ( x ) be¬ 
longing to the face Ph will be determined, on the grounds 
of (6) and (10), by the equalities 

t 

x i = ^2 PkZik where p k > 0. (k = 1,2, . . ., t; i = 1,2, . . ., m) 
k = 1 

( 12 ) 

Let us notice that all the points (11) verify the equa¬ 
tion 

Vh(x) = 0 (13) 

and satisfy the conditions 


Vk(x) > 0. (k = 1,2, . . . , a) 


One obtains thus the equalities 


VhUr)> 0 (r = 1,2, . . . , t; k = 1,2, . . . , a) (14) 


The face Ph being in m-1 dimensions, the equalities 
(14) would define the coefficients of the equation (13) to 
a close by common factor. 


Let us suppose that one had determined this way all 
the faces 

Pl,P 2 ,...,Pa (15) 

in m-1 dimensions of the domain R. 

Let us suppose that the points 

(€fc) = (€ifc,€ 2 fc, €mfc) (fc = 1,2,. . ., s) (16) 

characterise the various edges of the domain R belonging 
to the various faces (15). 

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G. F. Voronoi (1908) 


By indicating 

S 

x i = ^2 pk^ik where p k > 0, (k = 1, 2, . . . , s; i = 1,2, . . ., m) 
k = 1 

(17) 

one obtains a set of points which all belong to the do¬ 
main R. 

I say that any point ( x ) belonging to the domain R 
can be determined with the help of equalities (17). 

One can suppose that the point ( x ) does not belong 
to any one of the faces (15), since any point belonging 
to them can be determined with the help of equalities 
( 12 ). 

By supposing that one had the inequalities 

Vk(x) > 0, (k = 1,2, . . . , a) 

let us arbitrarily choose a point (£ r ) among those of the 
series (16) and let us admit 

x'i = Xi — p^i r where p > 0. (i = 1 , 2 ,..., m) (18) 


So long as the parameter p is sufficiently small, one 
will also have 


Vk(x') > 0 . (k = 1,2,. . ., < t ) 


By making the parameter increase in a continuous 
manner, one will determine with the help of equalities 
(18) a point ( x ') verifying an equation 

Vh(x') = 0 

and satisfying the condition 

Vk(x') > 0 . (k = 1 , 2 ,. . ., < t ) 


The point obtained ( x ') belongs to the face Ph, there¬ 
fore one can declare 

t 

x'i = Yj p' k £,ik where p' k > 0. (k = 1,2, . . . , f; i = 1,2, . . . , m) 
k = 1 


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G. F. Voronoi (1908) 


Monograph Translations Series 


By virtue of (18), it becomes 

t 

Xi = p£,ir + ^2 P'kZik where p > 0, p' k > 0. 

k =1 

(k = 1,2, ... ,t-,i = 1,2, ... ,m) 

It remains to demonstrate that the domain F is in m 
dimensions. 

Let us notice that all the points determined by the 
equalities (17) with condition 

Pk > 0 (k = 1,2, . . ., s) 
are interior to the domain R. 

In effect, all the points (16) verify the inequalities 
yn^k)> o. (k= 1,2.s; fa = 1,2,...,a) (19) 


By multiplying these inequalities by pk, let us make 
the sum of inequalities obtained; one will have, because 
of (17), 


Vh(x) = ^2 PkVh(^k) > o. (h = 1,2, . . . , a) 
k = 1 


By virtue of (19), one will have the inequality 

y h (x) > 0, (h = 1, 2, . . ., s) (20) 

so long as the numbers 2 /h(£i), ^(£2), • • • , yh(Cs) do not 
cancel each other out. 

One can not suppose that the equalities 
yn{£,k) = 0 (k = 1,2, . . ., s) 
holds, because otherwise all the equations 

Vi(x) = 0, 2/2(2:) = 0,. . ., y CT ( x) = 0 

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G. F. Voronoi (1908) 


would be of proportional coefficients, which is contrary 
to the hypothesis; therefore one will have the inequali¬ 
ties (20), and it follows that the domain R is of m di¬ 
mensions. 

We have demonstrated that the condition (£) is suf¬ 
ficient for the domain R to be of m dimensions. It is 
easy to demonstrate that this condition is necessary. 


We have defined in Number 10 the faces in m — 1 
dimensions of the domain R. This definition can be gen¬ 
eralised. 


Definition. One will call face in p dimensions of the do¬ 
main R (p = 1,2, . . ., m — 1) a domain P(p) formed from 
points belonging to the domain R and verifying a system 
of equations 


y k (x) = 0, (k = 1,2,...,r) (21) 

provided that these equations define a domain in p di¬ 
mensions composed of points which, all, do not verify any 
other equation y T + i(x) = 0, . . ., y a (x) = 0. 


Let us choose among the points (16) all those which 
verify the equations (21). 

By indicating with 

f,k — (f,lk j f,2k j • • • j f,mk) j (k — 1,2, . . . , t) 

one will declare 

t 

Xj = Y] Pkf,ik where p k > 0. (k = 1,2,... ,t\i = 1,2, ... ,m) 

k = 1 

( 22 ) 

It is easy to demonstrate that any point ( x ) belong¬ 
ing to the face P(p) can be determined with the help of 
equalities (22). 


32 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


Corollary. Each face of the domain R is a set of points 
determined by the equalities (22) provided that any point 
belonging to it could not be determined by the equalities 

S 

x i = ^2 Pk^ik where p k > 0, 
k = 1 

(k = 1,2 ,,s m ,i = 1,2 ,,m) 

unless all the parameters pt+i , Pt+ 2 , ■ ■ ■ , p s do not cancel 
each other. 


Any point belonging to the domain R either is in¬ 
terior to the domain R or is interior to a face of that 
domain. 

Let us suppose that the point ( x ) be interior to a 
face P(p) of the domain R which is formed from all the 
points determined by the equalities (22). 

I argue that one can always determine the point (x) 
by the equalities (22) provided that 

Pk > 0. (k = 1,2, . . ., t) 


To demonstrate this, let us indicate 

t 

Pi — ^ ' f,ik- (i — 1 ) 2 ,..., m ) 

k = 1 


The point (a) is interior to the face P(p). 

By admitting 

x\ = Xi — poii where; p > 0, (i = 1,2m) (23) 

one obtains a point (x ' i ) which will be interior to the 
face P(y) so long as the parameter p will be sufficiently 
small; it follows that 

t 

x'i = P'k&k where p’ k > 0. 
k = 1 


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G. F. Voronoi (1908) 


(k = 1,2, . . ., t; i = 1,2, . . ., m) 
By virtue of (23), one obtains 

t 

x i = *52 (p + p'k)€ik, (i = 1, 2 > • • • > m) 
k =i 

and by making 

P + Pk = Pk, (k = 1,2, . . ., t) 

one will have 

t 

Xi = Pkiik where p k > 0. 
k = 1 

(k = 1,2, . . . ,t; i = 1, 2, . . ., m) 


Let us notice that by making p = m and t = s, one 
will indicate with the symbol P(m ) the domain /?; one 
concludes that any point ( x ) which is interior to the do¬ 
main R can be determined by the equalities 

S 

Xi = ^2 PkZik where p k > 0. 
k = 1 

(k = 1,2, ... ,s\i = 1,2, ... ,m) 


On the correlative domains. 


Definition. Let us suppose that a domain R be deter¬ 
mined with the help of inequalities 

PlkXl + P2kX2 + • • • + PmkXm >0. (fc = 1,2 , . . . , tj) 


One will call correlative to the domain R the domain 
1Z which is formed from all the points ( x ) determined by 
the equalities 

<7 

Xi = ^2 PkPik where p k > 0. (k = 1,2, . . ., a\ i = 1 , 2, . . ., m) 
k= 1 

( 1 ) 

34 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


I say that the domain 1Z will be in m dimensions, 
if the domain R does not possess points verifying the 
equations 

PlkXl + PlkX 2 + PmkX m = 0, ( fc = 1, 2, . . . , (T) 

the point x\ = 0, X 2 = 0,. . ., x m = 0 being excluded. 

In effect, if all the points of the domain 1Z verified 
the same equation 


+ £ 2X2 + • • • + £m,Xm = 0 , 

one would have the equalities 

flPlk + f,2P2k + • • • + f,mPmk =0, ( fe = 1, 2, . . . , £7) 
by virtue of (1), which is contrary to the hypothesis. 


Theorem. By supposing that the domain R be formed 
from all the points (x) determined by the equalities 

S 

aq = ^ Pkf,ik where p k > 0, (k = 1, 2, . . . , s; i = 1,2, . . ., m) 

fc=i 

( 2 ) 

one will define the correlative domain 1Z with the help of 
inequalities. 

flkXl f,2kX2 + • • • + f,mkX rn Z 0- (k — 1,2,..., s) (3) 


Let us indicate by 7 Z 1 the domain determined with the 
help of inequalities (3). 

On the ground of the supposition made, all the points 

(£ll, £21, ■ ■ ■ , £m.l), (€l2, £22, ■■■, £m2 ),■■■, (£l s 5 f, 2 s 7 ■ ■ ■ 7 f,ms ) 

characterise the edges of the domain R, and one will have 
the inequalities 

Plhf,lk d - P2hf,2k Pmhf,mk 

God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 35 



Monograph Translations Series 


G. F. Voronoi (1908) 


(h= l,2,...,a;h= 1,2,(4) 


We have seen in Number 10 that each face Ph in m — 
1 dimensions of the domain R is characterised by the 
points 

(£l 1) £21 j • • • j 1 ) ) (£l2 j £22 j • • • 1 ( 1112)1 • • • ) (£ It > ^2t) • • • > t) 

which verify the equation 

Vh = 0 (5) 

of the face Ph- One obtains the equalities 

Plhlilk P P2h£,2k + • • • + Pmh£,mk = 0 (fc=l,2,...,t) 

which define the coefficients pm,P 2 h, ■ ■ ■ ,Pmh of the equa¬ 
tion (5) to an immediate common factor. 

One concludes, by virtue of the definition established 
in Number 9, that the point (pih,P 2 h, ■ ■ ■ ,Pmh) charac¬ 
terises an edge of the domain 7Z'. 

By attributing with the indice h the values 1, 2, . . . , a, 
one obtains a series 


(Pll>P21, • • • j P ml) i (Pl2,P22, • • • j Pm 2) j • • • , (Pier , P2cr, • • • , Pmcr) 

of points which characterise different edges of the do¬ 
main 1Z'. 

I argue that the domain 1ZJ does not possess other 
edges. To demonstrate this, let us suppose that a point 
Pi, p 2 , • • • , Pm characterises an edge of the domain 1Z'. 

One will have the equalities 

Pl^lh + P2^2h + • • • + Pm^mh = 0, (h = 1,2, . . . , t) (6) 

which define the coefficients Pi,P 2 i ■ ■ ■ iPm to a nearby 
common factor, and one will have the inequalities 


PlClh + P2^,2h + • • • + P rri£,mh ^ 0. (fc=l J 2,...,s) (7) 

36 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


Let ( x ) be any point of the domain R. One will de¬ 
termine the point ( x ) with the help of equalities (2). By 
multiplying the inequalities (7) with p/, and by making 
the sum of inequalities obtained, one will have, because 
of (2), 

PlX± + P 2 X 2 + • • • + Pm^Cm ^ 0 • 


One concludes that the inequalities 

- PiX 1 - P 2 X 2 - ■ ■ ■ - PmXm > 0 

and 

PlkXl + P2kX 2 + • • • + PmkXm — 0 ; 

define a domain which is not in m dimensions. 


By virtue of the fundamental theorem of Number 10, 
one will determine in this case positive values or zeros 
of parameters p,p\,...,p a which reduce the equation 

~p(pixi + P2X2 + • • • + Pm^m )+ 

cr 

^2 Pk (PlkXl + P2kX2 + • • • + PmkXm ) = 0 
k = 1 

into an identity. 


It follows that 


Pi = V] — Pik where — > 0.44 

ti P P 


(k = 1,2,. . ., ct; i = 1,2, . . ., m) 


By substituting (6), one will have 


^ ' ( S,±hP±k S,2hP2k £mhPmk) — 0. 

k = 1 P 


(h = 1,2,. ..,t) 


By virtue of (4), one finds 


Pk 


(£,lhPlk + £,2hP2k + •••+€ mhPmk ) — 0* 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 37 



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G. F. Voronoi (1908) 


(h — 1,2 , }t', k — 1,2,.,.,cr) 

Let us suppose that > 0, then 

f,lhPlk f,2hP2k “H • • • “1“ £,mhPmk — 0> (hi — 2, . . . , t) 

therefore the coefficients Pi, P 2 , ■ ■ ■, Pm, by virtue of (6), 
are proportional to the coefficients pik, P 2 k, ■ ■ ■ , Pmk] it 
follows that the points (pik, P 2 k, ■ ■ ■ , Pmk) and (pi,P 2 , ■ ■ ■ , 
Pm) characterise the same edge of the domain 1Z'. 

By virtue of the fundamental theorem in Number 10, 
all the points of the domain 7 Z' will be determined by the 
equality (1), this results in that the domains 1Z and 7 Z' 
coincide. 


Corollary. Let us suppose that a face P(p) in p dimen¬ 
sions of the domain R be determined by the equations 

PlkXl + P2kX 2 + • • • + PmkXm = 0, ( k = 1,2, . . . , T) 

and that any point ( x ) belonging to the face P(p) be de- 
termined by the equalities 

t 

Xi = ^2 Pkf,ik where p k > 0. 
k = 1 

(k = 1, 2 , . . . , t; i = 1, 2 , . . . , m) 


The correlative domain 1Z will possess a correspond¬ 
ing face J3(m — p) in m — p dimensions determined by the 
equations 

ftlkX 1 f f 2kX 2 “H • • • “H f,mkXm — 0 (^ — 1,2, . . . , t) 

and any point ( x ) belonging to the face B(m — p) will be 
determined by the equalities 

T 

Xi = ^2 pkPik where p k > 0. 
k = 1 

(k = 1,2, . . ., t; i = 1,2, . . ., m) 


38 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


Definition of domains of quadratic forms correspond¬ 
ing to the various perfect forms 


Let us consider any one perfect quadratic form cp. 

Let us suppose that all the representations of the 
minimum of the form <p make up the series 

(hi, hi, ■ ■ ■ , Ini') , (l 12, 122, ■ ■ ■ ,ln2), ■ ■ ■ , (hsjhs, ■ ■ ■ ,lns )• (1) 

By indicating 

A k — 11 k X 1 “1“ 12 k X 2 • • • “1“ l nkX n , (k — ( 2 ) 

one corresponds to the series (1) a series of linear forms 

Ai, A 2 , . . ., A s . 


Let us consider a domain R of quadratic forms de¬ 
termined by the equality 

S 

f(xi,x 2 ,. . . ,x n ) = Pk^l 

k=l 


with condition that 


Pk > 0. (k = 1,2, . . ., s) 


One will say that the domain R correspond to the 
perfect form <p. 

Let us notice that the domain R is in n ( n + 1 '> dimen¬ 
sions. 

By supposing the contrary let us suppose that all 
the quadratic forms belonging to the domain R verifies 
a linear equation 

^(/) = Y Pi 3 ai 3 = °' 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 200 7 39 



Monograph Translations Series G. F. Voronoi (1908) 

On the ground of the established definition, one will 
have the equalities 

^(A|) = 0 (k= 1,2,....a) 

or, that which comes to the same thing, because of (2), 
Pijlikljk — d ( k — 1 j 2, . . . , S ) 
which is impossible, the form ip being perfect. 

On the ground of what has been said in Number 9— 
14, the domain R possesses s edges characterised by the 
quadratic forms 

A?, A3,..., Aj. (3) 


Let us suppose that one had determined all the faces 

Pi, F > 2, ■ ■ ■ , P a 

in n< , n + 1 '> — i dimensions of the domain R. 

Each face Pk can be determined by two methods: 

1. All the quadratic forms belonging to the face Pk 
verify an equation 

^k(f) = = 0 

which can be determined in such a way that the inequal¬ 
ity 

**(/)> 0 

held so long as the form / belonging to the domain R is 
exterior to the face Pk- 

2. By choosing among the quadratic forms (3) these 

\2 \2 \2 

Ai, A 2 , • • • , A t 

which verify the equation (4), one will determine all the 
quadratic forms belonging to the face Pk by the equali¬ 
ties 

t 

f(x !,X 2 , ■ ■ ■ ,X n ) = ^PkXl, 
k = 1 


40 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 
where 


Monograph Translations Series 


Pk> 0. (fc = 1,2,....t) 


By virtue of the theorem of Number 14, the domain 
R can be considered as a set of points verifying the in¬ 
equalities 

**(/)> 0. (k=l,2,...,a) 


On the extreme quadratic forms 


Let us indicate by M(aij ) the minimum and by D(aij) 
the determinant of a positive quadratic form Jj aijXiXj ■ 


The positive quadratic form 


^/.D (aij ) 

determinant 1 and will possess the minimum 


J2 a ij x i x j will be of 


M (a jj ) 

'{/ D (a t -j ) 


M(aij). 


Let us examine the various value of the function 

M(aij) 

which is well determined in the set (/) of all the positive 
quadratic forms in n variables. 


Definition. One will call extreme J a positive quadratic 
form aijXiXj which enjoys the property that the corre¬ 
sponding value of the function JA(aij ) is minimum. 


Let us notice that the function M(aij) does not ch¬ 
ange its value when one replaces quadratic form Jj aijXiXj 
by a form of proportional coefficients. 

By attributing to the coefficients of the extreme form 
Jj aijXiXj variations 

e ij = e ji (* = 1,2,. . ., n ; j = 1,2, . . ., n) 

| See the Memoire of Mr. ’s Korkine and Zolotareff, Sur 
les formes quadratiques [On the quadratic forms], Math- 
ematische Annalen V. VI, p. 368 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 jl 



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G. F. Voronoi (1908) 


satisfying the condition 

I j I <- e , (* — 1,2, . . ., n , j — 1,2,. . . , n) (1) 

e being an arbitrary positive parameter, let us examine 
the corresponding value of the function M(aij). 

On the ground of the definition established, one can 
determine the parameter e such that the inequality 


M(a,ij + tij) < M(aij) ( 2 ) 

held with condition (1) and so long as the coefficients 
eij are not proportional to the coefficients 

aij. (i = 1,2, . . . ,ra; j = 1,2, . . ., n) 


Theorem. For a quadratic form QijXiXj to be extreme, 
it is necessary and sufficient that it be perfect and that 
its adjoint form be interior to the domain 

corresponding to the form JfaijXiXj. 


Let us indicate by 

Oil, hi, ■ ■ ■ , In l), 0l2, 1 22, • • • 0n2), • • • , (l Is, hs, ■ ■ ■ , l ns) (3) 

the various representations of the minimum M(aij) of the 
form JfaijXiXj. 

Let us consider a quadratic form + peij)xiXj , 

the parameter p being arbitrary. One can determine an 
interval 

— 5 < p < 6 where 0 < <5 < 1 (4) 

such that all the representations of the minimum of the 
form J2(aij + peij)xiXj are found among the systems (3) 
so long as the variations tij satisfy the condition (1). 

By indicating with 

AT — f ' (a l y “I - peij)likljk and AT — f a j j l j r. I j k (3) 

f2 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


the minima of forms Yl( a ij + p€ij)xiXj and J2 a ij) x i x j and 
with D' and D their determinants, one will have 


+ peij) 




^D 7 


A4 (o-ij ) 


a ij likljk 


sfD 


By virtue of (2), one obtains the inequality 
“I - P^-ij)likljk ^ Edijlikljk 

r <fly </T> 

or, that which comes to the same thing, 


v- / D' 

P 2_ _J e ij likljk < FT W — 1 


D 


( 6 ) 


This declared, let us suppose that the form J2 a ij x i x j 
be not perfect. 

One will determine in this case the variations eij such 
that the equalities 


X ^ ^ijljkl jk — Oj 


held. By virtue of (6), one will obtain the inequality 

D' > D. 


By developing the determinant D' into a series, one 
will have the inequality 


dD 


The parameter p being arbitrary satisfying the con¬ 
dition (4), it is necessary tht 


E 


dD 

~ ij da 




= 0. 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 f3 



Monograph Translations Series 


G. F. Voronoi (1908) 


Mr.’s Korkine and Zolotareff have demonstrated J 
that in this case one will always have the inequality 


E 




d 2 D 

daij dcikh 


< 0, 


therefore the inequality (7) is impossible. 


We have demonstrated that the form c p = Y) a ij x i x j 
has to be perfect. 

Let us suppose that the domain R corresponding to 
the perfect form c p be determined by a inequalities 

> °. (r = 1,2, . . . , a) 


On the grounds of these inequalities, one will have 

^r(A^) = ^P^likljk >0. (k = 1,2, . . ., s; r = 1, 2, . . ., a) 

( 8 ) 

Let us declare 

eij = tp\ r ^ where t>0. (i = 1, 2 , . . ., n; j = 1, 2 , . . ., n) 
By virtue of (6), one will have 

P* ^2 p i r j )likl l k < M ^ ^ • ( 9 ) 


Let us attribute to the parameter p a positive value 
satisfying the condition (4), by virtue of (8) and (9) 
there will arrive 


D 1 > D. 


By developing the determinant D' into a series, one 
obtains the inequality 




dD 


( r ) 

^ da 


+ 


(pty 


E (r) (r) 

PijPih- 




d 2 D 


daij dakfi 


+ ... > 0 . 


| Mathematische Annalen, V. XI, p. 250 


44 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


The positive parameter p being as small as one wish, 
it follows that 


I>S’ 


dD 

dan 


> 0 . 


(r = 1,2, . . . , a) 


It is thus demonstrated that the form J2 ~§^~ x i x j > ad- 
joint to the form ip, is interior to the domain R. 

I argue that in this case the perfect form p> will be 
extreme. 

By supposing the contrary, let us suppose that the 
inequality 

-)- eij) > Af(ajj) (10) 

be verified by any one system of variations (i = 1,2, 

. . . , n; j = 1, 2 , . . . , n) satisfying the condition (1) however 
small the parameter e may be. 

By virtue of (10), one obtains 

( n rw \ 

^tijlikljk > M I y—- 1 ; (fe = 1,2, . . ., s) (11) 

the inequality obtained has to hold whatever may te the 
value of the index k = 1, 2, . . ., s. 

By indicating 



(z — 1, 2, . . . , n ; j — 1,2, . . . , n) (12) 

let us examine the quadratic form 

ip 0 (x 1 , x 2 , ■ ■ ■ , X n ) =^2(aij + T] ij )xiX j . (13) 


By virtue of (12) the form <p 0 is of determinant D. 

By choosing the parameter e sufficiently small, one 
can suppose that 

\Vij \ < r), (i = 1, 2, . . . , n; j = 1,2, . . . , n) (14) 

God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 45 



Monograph Translations Series G. F. Voronoi (1908) 

r) being a positive parameter as small as one would like. 


By virtue of (5), (11) and (12), one obtains 

y^Vijlikljk > 0- (fc = 1,2, . . ., s) (15) 


By developing the determinant D of the form (13) 
into series, one will find 


D + l >« 5 ^- + h » = d ' 


(16) 


In this equality the remainder R 2 verifies an inequal¬ 
ity 

|i?2| < tfP, 

P being a positive number not depending on the param¬ 
eter r] so long as rj < 1. 

By virtue of (16), one obtains 

< V 2 P- (17) 


E 7 ^ 


dD 


dai 


We have suppose that the quadratic form -§P-XiXj, 
adjoint to the form ip, be interior to the domain R. On 
the ground of that which has been said in Number 13, 
one will determine the form J2 x i x j with the help of 
the equality 


E 


dD 

- - XiXj 

u a ij 


E ? kX * 

k — 1 


(18) 


where 

Pk > 0. (k = 1, 2, . . . , s) (19) 


The equality (18) can be replaced by the following ones: 


0D_ 

d a ij 


S 

^ ^ Pk likljk- (^ 1 ^ , . . . , 71 , j — 1,2, . . • , Tl ) 

k — 1 


46 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


By multiplying these equations by 77 ^ and by adding 
up the equalities obtained, one will have 




dD 

O&ij 


^ ' Pk ^ ' Pi j likl jk • 
k = 1 


( 20 ) 


By virtue of (15), (17) and (19), one obtains the in¬ 
equalities 

_ p 

0 ) ( Pijlikljk ^ P 5 (A) — l,2,...,s) 

Pk 


therefore one can admit 

) ' Pijlikljk — T~kP j (k — 1 j 2,_..., S ) (21) 

and the positive numbers or zeros Tk (k = 1,2 , . . . , s) will 
not exceed fixed limits which do not depend on the pa¬ 
rameter 77 . 

After the definition of perfect forms, the equations 
(21) admit only a single system of solutions. By effecting 
this solution of equations ( 21 ), one obtains 

Pij = Tijp 2 (i = 1, 2, . . ., n; j = 1,2, . . ., n) 

where 

I T ij I < T, ( 7 = 1, 2, . . . , ra; j = 1,2,..., 71 ) 

T being a positive number which does not depend on the 
parameter 77 ; therefore one will have the inequalities 

\pij\<p 2 T. (i = 1,2, . . . , ra; j = 1, 2, . . . , ra) (22) 


This stated, let us take any one positive fraction 1 ? 
and declare 



By virtue of (14), one will have 


\P 


I "S' rj-i 5 


( 7 = 1,2,... ,n; j = 1,2,...,ti) 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 f7 



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G. F. Voronoi (1908) 


and because of (22), it will become 


0 ' 


\Vij\ < -rjT- (* = 1,2, . . ., n; j = 1,2, . . ., n) 


By admitting 

V 2 

tiij < p 1 

one will have, because of (22), 

i ? 4 

\Vij\ < ~p, (i = 1,2, ■ ■ ■, n; j = 1, 2, . . . , n) 
and so on. 

One will obtain in this manner the inequalities 

0 2k 

\Vij \ < ~p~ ( l= 1 ,2, . . ., n; j = 1, 2, . . . , ra; fc = 0, 1,2, . . .) 

it follows that 

Vij = 0. (i = 1,2, . . ., n; j = 1,2, . . ., n) 


By virtue of (12), one obtains 



therefore the coefficients are proportional to the co¬ 
efficients ciij (i = 1,2, . . . , n; j = 1, 2, . . . , n), which is con¬ 
trary to the hypothesis. 

Properties of the set of domains corresponding to the 
various perfect forms in n variables. 


Any perfect form cp will be transformed by an equiv¬ 
alent perfect form p 1 with the help of any substitution S 
of integer coefficients and of determinant ±1. 

Let us indicate by R and R' the domains correspond¬ 
ing to the perfect forms <p and ip' and by T the substitu¬ 
tion adjoint to the substitution S. 


48 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


One will easily demonstrate that the domain R will 
be transformed into an equivalent domain R' with the 
help of the substitution T. 

One concludes that there exists a finite number of 
domains equivalent to the domain R. 

Let us indicate by (f?) the set of all the domains cor¬ 
responding to the various perfect forms in n variables. 

The set (f?) can be divided into classes of equivalent 
domains. 

On the ground of that which has been previously 
said, the number of different classes of the set (f?) is 
equal to the number of classes of perfect forms in n vari¬ 
ables. 


Theorem. Let us suppose that a quadratic form f be 
interior to a face Pin) in p dimensions of the domain 
R(p= 


The form f will belong only to the domains of the set 
( R ) which are contiguous through the face P(p). 

Let us suppose that the domain R be characterised 
by the quadratic form 


A?, A|.(1) 

and that the face P(p) in p dimensions of the domain R 
be characterised by the quadratic forms 

(2) 


In the case p = n dH-i) , one w jh admit t = s, and the 
symbol P ^ n ( n + 1 '> ^ will indicate the domain R. 


The quadratic form / being interior to the face P(p ), 
God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 



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G. F. Voronoi (1908) 


one will have the equality 


t 

f(xi,x 2 , . . ., X n ) = Pk^l where p k > 0. (k = 
k = 1 


1,2 

(3) 


Let us suppose that the same form / belonged to 
another domain R' of the set ( R ). 


Let us suppose that the domain R' be characterised 
by the quadratic forms 


\' 2 \' 2 v 2 


(4) 


and that the form / be interior to the face P'(u') of the 
domain R’ characterised by the quadratic forms 


\' 2 V 2 
A i , a 2 , 


A' 


(5) 


One will have, on the ground of the supposition 
made, 


f(xi,x 2 , • • • , x n ) = ^ Ph x 'h 2 where p' h > 0. (h = 1,2, . . . , r) 
h= 1 

( 6 ) 

This declared, let us indicate by <p and ip 1 the perfect 
forms corresponding to the domains R and R' and sup¬ 
pose, for more simplicity, that the minimum of forms <p 
and </?' be M. 


By indicating with the symbol (/,/') the result 

(/, /') = Y2 ai i a 'ij > 

from two quadratic forms 

f (.X 1 , 3? 2 , • • ■ , X n ) — ^ ' Q'ij X iX j 


and 


f(x i, x 2 , . . . , x n ) = ^2 aljXiXj , 

let us examine two results and (/,<£>'). 

50 February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


By virtue of (13), one obtains 

t t 

(/, V) = Y Pk{T>, and (/, if') = Y Pk(<p', A£ 

fc=1 /c=l 


By virtue of (6), one obtains 


(/>¥>) = Y PhiV’Xh 2 ) and (/>¥>') = Yp'hW' X 'h 

h=l h=l 


Let us notice that 

( <p,\l) = M and ( cp', X 2 k ) > M ; (fc = 1,2,..., 
(v,\' h 2 )>M and (^',A' h 2 ) = M. (fc = 1,2, . . ., ct 


From equalities (7), one derives 

(/, ¥>') - (/, V) = Y P k [(^'> A fc) _ (V. A fc)l > 

k = l 

and by virtue of (3) and (9) there comes 


(/, ¥>') - (/, ¥>) > 0. 


From equalities (8), one derives 

(/. ¥>') - (/. ¥>) = Y Ph [(¥>'. A 'h 2 ) - (V. A 'h 2 )] . 

h = l 

and by virtue of (6) and (10), one will have 


(/, ¥>') - (/, ¥>) < 0. 


It follows that 


(/, ¥>') = (/, V 3 ), 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 


( 7 ) 


)• ( 8 ) 

) ( 9 ) 

( 10 ) 

( 11 ) 


( 12 ) 


2007 51 



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G. F. Voronoi (1908) 


and the equalities (11) and (12) give 

(¥>', Afc) = (<p, X 2 k ), (k = 1,2, . . .,t) 
= (V,K 2 )- (h= 1,2,... j t) 


By virtue of (9) and (10), there arrive 
(<p,\' h 2 ) = M, (h = l,2 J ... J r) 
(V',A l) = M. (k = 1,2,...,*) 


(13) 

(14) 


By virtue of equalities (13), the quadratic forms (5) 
are found among those of the series (1). By virtue of 
(14), the quadratic forms (2) are found among those of 
the series (4). 

I argue that in this case the series (2) and (5) contain 
the same forms. 

To demonstrate this, let us suppose that all the forms 
belonging to the face P(/x) verify the equations 

*l(/) = 0, * 2 (/) = 0,-.., * r (f) = 0 

and that any form belonging to the domain R verifies 
the inequalities 

*i(/)> 0, * 2 (/)>0,..., *r(/)>0. (15) 

By virtue of (6), one will have 

p'^iiX 1 ! 2 ) + p' 2 *i(\' 2 2 ) + ■ ■ ■ + p't(K 2 ) = 0, (*= l,2,...,r) 

and because of (15), one finds 

'MAk 2 ) = 0; (* == 1,2,.. .,r;h = 1,2,. . . , r) 

therefore all the forms of the series (5) belong to the 
series (2). 

In the same way, one will demonstrate that all the 
forms of the series (2) belong to the series (5). 

One concludes that the faces B(p.) and P'(v) coincide, 
therefore the domains R and R' are contiguous through 
the face P(p,). 


52 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


Corollary. A quadratic form which is interior to a do¬ 
main of the set ( R ) can not belong to any other domain 
of that set. 


21 


Theorem. Let us suppose that to a face P(y) of the do¬ 
main R belong positive quadratic forms. In this case, the 
number of domains of the set ( R ) contiguous through the 
face P(j- 1 ) is finite. 


Let us indicate by 


R, Ri, i?2, • • • 

the domains of the set (f?) contiguous through the face 
P(/x). Let 

¥b <Pl,<P2, ■ ■ ■ 

be the corresponding perfect forms having the minimum 

M. 


On the ground of the supposition made, one positive 
quadratic form / will be interior to the face P(p). 

We have demonstrated in the previous number that 

(/,¥>) = (/,¥>i) = (/, ¥> 2 ) = •■ ■ (16) 


It is easy to demonstrate that the number of perfect 
forms having the minimum M and verifying the equali¬ 
ties (16) is finite. 


Algorithm for the search for the domain of the set (R ) 
contiguous to another domain by a face in n ^ n 2 +1 ' ) — 1 di¬ 
mensions 


Let 


(p (x 1, X2 , . . . , X 77 , ) — Oij X}X 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 53 



Monograph Translations Series 


G. F. Voronoi (1908) 


be a perfect form having the minimum M the various 
representation of which make up a series 

(^11, l2h ■ • • 0l2, 1 22, • • • , 1 712 ), ■ ■ ■ , (l Is, 12s, ■ ■ ■ , Ins)- (1) 


Let us suppose that a face P of the domain R cor¬ 
responding to the perfect form ip be determined by the 
equation 

*(/) = J2Pij a H = 0 

and by the condition 


*(/) > o 

which is verified by any quadratic form belonging to the 
domain R. 

Let us suppose that the face P be characterised by 
the quadratic forms 


\2 \ 2 w 

Ai, A 2 , • • • , A t 


( 2 ) 


where 


k — 1 1 k -E 1 12k -E2 “ 1 “ • • • “ 1 “ lnk-E n • ( h — 1 , 2 ,. . . , S ) 


On the ground of the supposition made, one will have 
the equalities 


^ Pijhkljk = 0 (k = 1,2, . . . , t) (3) 

which define the coefficients Pij (i = l,2,...,n;j = 1,2, 
. . . , n) to an immediate common factor. 

Let us suppose that the face P could belong to the 
other domains of the set (R). Let us indicate by R' a 
similar domain. Let ip' be the perfect form corresponding 
to the domain R'. 

By virtue of the supposition made, the quadratic 
form (2) belong to the domains R and R'. It results in 
that the systems 

(In, 121, ■ ■ ■ , Ini), (l 12, 122, [• • • , ] ln2 ) , ■ ■ ■ , (l It, 12t, ■ ■ ■ , Int) (4) 


54 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


corresponding to the forms (2) represent the minimum 
of forms ip and ip 1 . 

Let us suppose, for more simplicity, that the forms ip 
and ip' had the minimum M. One will have the equalities 

^ ' ciijlikljk —— Mi and ^ ' &ijlikljk —— Ml , (k — 1, 2,. . ., t) (5) 
by putting 

<^'(^ 1 ,^ 2 , • • • , x n ) = ^aGxiXj. 


From equation (5), one gets 

( a ij — a ij)likl jk = o, (k = 1,2, . . . , t ) 

and by virtue of (3), it becomes 

a 'ij = a ij + PPij- (i = 1,2, . . ., n; j = 1,2, . . ., n) (6) 

Let us indicate 

<F(x 1 ,x 2 , ■ ■ ■ ,x n ) = Y^Pij x i x j- 


By virtue of (16), one obtains 
(p'(x 1 ,x 2 , ■ ■ ■, x n ) = cp(x 1 ,x 2 , ■ ■ ■ ,x n ) + p'&(x 1 ,x 2 , ■ ■ ■ ,x n ). 

( 7 ) 

This stated, let us choose in the series (1) a system 
(Zi h, l 2 h, • • •, l n h ) which does not belong to the series (4). 
As 


’F ( Z lh j • • • j lnh) — Af </? {llhi l 2 ft, j • • • j lnh ) ^ Af 

and 

(l lh i 12h i ••• i lnh) ^ 0, 
one deduces from the equality 

SP (llh : l2hi ■ ■ ■ : l nh) = SP (Z lh > 12h :•••■> l nh) pAt (l lh , 12h > • ■ ■ > Z nft.) 

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Monograph Translations Series 


G. F. Voronoi (1908) 


the inequality 


p > 0. 


The supposition p = 0 being obviously impossible, one 
obtains 

P> 0 , 


and it follows that 


i P (l lh i l2h i • • • i Inh ) ^ Ml . 


Let us indicate by 

(I'l , *2, ■ ■ ■ , O. (M. ■ ■ ■ . C), ■ - ■ , (8) 

all the representations of the minimum of the perfect 
form ip' which are not found in the series (4). By virtue 
of (7), one will have 

cp'(l[ k \l {k \...,l^) = 

cp(l[ k) ,l {k \ ■■■,l (k) ) + p^(l {k \l {k \. . . , 

( k= 1,2,....r) 

which results in 

<p(l[ k) ,li k) ,.. .,lW) > M and *(l[ k) ,l ( 2 k) ,...,lW) <0. 

(k= l,2,...,r) (9) 


The value of the parameter p will have for expression 
_ cp(l{ k \l {k \...,l { n k) )-l 




(*)■ 


(k 


1 , 2 , 


■ 


Let us examine any one value of the function 


v(x 1,X 2 , • • •, X n ) - M 
A'(x 1 ,X 2 , • • • , X n ) 


( 10 ) 


determined with the condition 


'&(x 1 ,x 2 ,. . . , x n ) < 0 . 


( 11 ) 


56 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


I argue that one will have the inequality 

<p(xi,x 2 , ■ ■ ■ , x n ) -M 

-/-\- P ' 

W(Xi,X2, ■ ■ ■ ,X n ) ~ 


Let us suppose the contrary. By supposing that 

<p(x 1,X 2 , • • • , X n ) -M 

-iT7-\ < P, 

V(xi, X2, ■ ■ ■ , Xn) 

one will find, because of (11), 

<p(xi,x 2 , ■ ■ ■ ,Xn) + P'&(xi,X 2 , ■ ■ ■ ,Xn) < M 
or, that which comes to the same thing because of (7), 
<p'(x 1,X2, • • • , Xn) < M, 
which is contrary to the hypothesis. 

We have arrived at the following important result: 

There exists only a single domain R' contiguous to the 
domain R through the face P. The corresponding perfect 
form p 1 will be determined by the equality (7) provided 
that the parameter p presents the smallest positive value 
of the function (10). 

Let us notice that by virtue of (3) and (9), all the 
quadratic forms belonging to the domain R’ verify the 
inequality 

*(/)< 0 . 


One concludes that the domains R and R' are found 
from two opposite sides of the plane in n bH~ 1 ) _ ^ dimen¬ 
sions determined by the equation 

*(/) = o. 


The smallest positive value of the function (10) can 
be obtained with the help of operations the number of 
which is finite. 


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The whole problem is reduced to the preliminary stu¬ 
dy of a system (Zi, Z 2 , . . . , l n ) of integers verifying the in¬ 
equality 

'i'ili,h, ■ ■ ■ ,l n ) < 0 

and satisfying the condition that the quadratic form 
<Po(xi,x 2 , ■ ■ ■ ,x n ) = <p(x 1} x 2 , ■ ■ ■ ,x n ) + p 0 '&(x 1 ,x 2 , ■ ■ ■ ,x n ), 
where one has admitted 

(P (l 1 1 12 i • ■ • t In') — M 

be positive. 

One will determine in this case all the systems (aq, 
x 2 , ■ ■ ■ , x n ) of integers verifying the inequality 

<p(x 1 ,x 2 , ■ ■ ■ ,x n ) > M (12) 

the number of which is finite, and one will find among 
these systems all those which define the smallest value 
of the function (10). 

Let us indicate by R, as we have done in Number 2, 
the upper limit of values of the parameter p. 

The problem is reduced to the study of a system 
(Z!, l 2 , . . . , l n ) °f integers verifying the inequality 

‘P’ill, I 2 , ■ ■ ■ , In) + R'P (ll > I 2 , ■ ■ ■ j In) <. M. (13) 


It can turn out that the equation 

<p(x i,x 2 , ■ ■ ■ ,x n ) + R^>(x 1 ,x 2 , ■ ■ ■, x n ) 


0 


will be verified by integers, one will determine them with 
the help of equations 


3 V + 


dx 


dxi 


0. (i = 1, 2, . . . , n) 


In the case where these equations can not be verified 
by any one system of integers, one will study the values 
of linear forms 


Op _l_ R d £ 

dxi dxi 


(i = 1,2, ... ,n) 


58 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


and one will determine as many as one wish of the sys¬ 
tems of integers verifying the inequality (13). 

By supposing that a system of integers (Zi, l 2 , ■ ■ ■ , l n ) 
verifying the inequality (13) were determined, one can 
look for the smallest positive value p of the function (10) 
with the help of the following procedure. 

The inequality (12) can be put under the form 

cp(xi,x 2 , ■ ■ ■ ,x n ) (l - ^ [<p(x 1 ,x 2 , ■ ■ ■ ,x n ) + 

RA'(xi, X 2 , ■ • • , Xn)] < M, 

and as 

<p(xi,x 2 , ■ ■ ■ ,x n ) + R'F(x 1 ,x 2 ,. . ., x n ) > 0, 
it becomes 

tp(xi, X 2 , ■ ■ ■ , Xn) (1 - < M, 


or differently 


<p(xi, X 2 , ■ ■ ■ , Xn) < M 


R 

R — Po 


Among the systems of integers verifying this inequal¬ 
ity one will find all the systems (8) searched for. 

Algorithm for the search for the domain of the set (R ) 
to which belongs an arbitrary positive quadratic form. 


Theorem. Any positive quadratic form belongs to 
at least one domain of the set (R). 


Let 

f (x 1 j x 2j ... j x n ) — ) ( a%jxiXj 
be any one positive quadratic form. 

Let us choose any one domain R from the set (f?). 
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Monograph Translations Series G. F. Voronoi (1908) 

Let us suppose that the form /, did not belong to 
the domain (f?). 

In that case all the linear inequalities which defined 
the domain R will not be verified. Let us suppose that 
one had the inequality 

^(/) = < °’ (!) 

Let us indicate by f?i the domain contiguous to the 
domain R through the face in n ( n + 1 '> _ i dimensions de¬ 
termined by the equation 

*(/) = o. 

By indicating with <p and (pi the contiguous perfect 
forms corresponding to the domains R and R±, one will 
have, as we have seen in Number 22, 

(pi(x 1 ,X 2 , ■ ■ ■ ,X n ) = (p(x!,X 2 , ■ ■ ■ ,X n ) + p^'(x 1 ,X 2 , ■ ■ ■ ,X n ) 

(2) 

where p > 0 and SI/(;ei, x 2 , • • • , x n ) = Y)Pij x i x j- 

Let us examine two results and (f,<p i). By 

virtue of (2), one will have 

(/, ¥>i) = (/, ¥>) + P(/> ^), 

and as, because of (1), 

(/, = 'YjPijO'ij < °> 

it becomes 

(/> <P) > (/, <Pi)- 

Let us suppose that by proceeding in this manner 
one obtains a series of domains 

R, Ri, J?2j • • • • (3) 

By indicating with 

¥>, ¥>i, ¥>2, ■ ■ ■ (4) 

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G. F. Voronoi (1908) 


Monograph Translations Series 


the series of corresponding perfect forms, one will have 
the inequalities 

(/,¥>) > (f,T> 1 ) > (f,<P 2 ) > ••• 

so long as the form / did not belong to the domains (3). 

By noticing that all the perfect forms (4) possess the 
same minimum M , one will easily demonstrate that the 
number of perfect forms (4) verifying the integrality 

(/,¥>) < P 

is bounded, whatever may be the value of the positive 
parameter P. 

One concludes that the series of domains (3) will 
necessarily be terminated by a domain R m to which be¬ 
longed the form / considered. 


Study of a complete system of domains representing 
the various classes of the set ( R ). 


Let R be any one domain of the set (f?). Let us sup¬ 
pose that one had determined all the domains 

R, R±, f?2, • • •, R<t (1) 

contiguous to the domain R through the various faces in 
n ( n + 1 '> — 1 dimensions, then let us suppose that one had 
determined all the domains contiguous to the domains 
(1) and so on. 

I say that by proceeding in this manner one will come 
across any domain of the set ( R ) arbitrarily chosen. 

For example, if one wish to arrive at a domain , 
one will choose a positive quadratic form / which is inte¬ 
rior to the domain R and one will proceed as we have 
done in Number 24. One will determine this way a series 
of domains 

R,R',R",...,R M ,R W 


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G. F. Voronoi (1908) 


which are successively contiguous through faces in n ( n + 1 l 
— 1 dimensions. 

We have seen in Number 19 that the set ( R ) can be 
divided into classes of equivalent domains the number of 
which is finite. 

Let us find a system of domains representing the var¬ 
ious classes of the set ( R ). 

By starting from the domain R, we have determined 
all the domains 

R' 1) R 2 ; ■ ■ ■ j Rcr 

contiguous to the domain R. By not considering equiva¬ 
lent domains as being different, let us choose among the 
domain (1) those which are not one to one equivalent 
and are not equivalent to the domain R. Let us suppose 
that one had obtained the series 

R, Ri, R 2 , ■ ■ ■ , Rp~i (2) 

of domains which are not one to one equivalent. 

One will study in the same way the domains contigu¬ 
ous to the domains R, R±, R 2 , • • •, R^-i and one will extend 
the series (2) by adding to it new domains 

R/X 5 + ’ ’ ’ 5 Rv 1 

which are not one to one equivalent and are not equiva¬ 
lent to the domains (2). 

By proceeding in this way, one will always obtain a 
series 

R, R±, f?2, • • • j R t 1 (3) 

which enjoys the following property: the domain belong¬ 
ing to the series (3) are not one to one equivalent and 
all the domains contiguous to the domains (3) are equiv¬ 
alent to them. 

The series (3) obtained presents a complete system 
of representations of various classes of the set (f?). 


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G. F. Voronoi (1908) 


Monograph Translations Series 


The study of the series (3) can be facilitated partic¬ 
ularly by the help of substitutions which transform into 
itself the domains of the set (f?). 

Let us suppose that the domain R corresponding to 
a perfect form p be determined by the inequalities 

(fc = 1 , 2 , . . ., ct) 


By declaring 

^k(x !,x 2 , ■ ■ ■ ,x n ) = XiXj, (k = 1,2,... ,a) 

one will determine, as we have seen in Number 22, by 
the equalities 

Pk = P + Pk^k (fc = 1,2,..., <x) (4) 

a perfect forms px, P 2 , ■ ■ ■, Pa- One will call them con¬ 
tiguous to the perfect form p. 

Let us indicate by g the group of substitutions which 
do not change the perfect form p. 

The perfect forms p\, P 2 , ■ ■ •, Pa being well determin¬ 
ed by the perfect form p, one concludes that all the sub¬ 
stitutions of the group g will only permute the forms 
Pi, P 2, ■ ■ ■ ,Pa- 


By not considering as different the forms in propor¬ 
tional coefficients, one can say, by virtue of (4), that the 
group g will only permute the quadratic forms 

^i, A/ 2 , • • • , (5) 


Let us suppose that one had chosen in this series the 
forms 

'Ll, 'La, . . . , i (6) 

which enjoyed the following properties: each form of the 
series (5) will be transformed into a form of the series (6) 
with the help of a substitution belonging to the group 


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Monograph Translations Series G. F. Voronoi (1908) 

g, the forms (6) can not be transformed one to one with 
the help of substitutions of the group g. 

The perfect forms 

T>k = T> + Pk'&k (k = 1,2, . . . ,p-l) 

can replace the perfect forms (4), therefore one will de¬ 
termine only the values of parameters pi, p 2 , ■ ■ ■, p^-i- 

The corresponding domains 

Ri, R 2 , ■ ■ ■, R^-i 

can replace the domains (1). 

It can come to pass that among the domains R, R±, R 2 , 
. . . , R^ i are found equivalent domains, one will recognise 
this with the help of particular methods. 

On a reduction method of positive quadratic forms. 


Definition. One will call reduced any positive quadratic 
form belonging to any one domain 

R, Ri, R 2 , ■ ■ ■, R T l (1) 

of a complete system of representations of various classes 
of the set (f?). 


Let us suppose that one had determined all the sub¬ 
stitutions 

Si,S 2 ,...,S m (2) 

which transform the domains contiguous with the do¬ 
mains (1) through the faces in n O+ 1 ) _ ^ dimensions into 
these domains here. 

Let / be any one positive quadratic form which is 
not reduced. One will determine with the help of the 
algorithm shown in Number 24 a series of domains 

R,R',R",.. . ,f? (h) 


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G. F. Voronoi (1908) 


Monograph Translations Series 


successively contiguous. Let us suppose that the domain 
R (h) be the first one which does not belong to the series 
(!)• 


With the help of a substitution S' which is found 
among those of the series (2), one will transform the 
domain R^ into a domain R k belonging to the series 
(!)• 


By transforming the form / with the help of the sub¬ 
stitution S' into an equivalent form /', one will determine 
with the help of the form /' a new series of domains 


Rk > R'k ! ■ ■ • j R 


(t) 


and so on. 

One will determine in this way a series of substitu¬ 
tions 

S',S",...,S^ 

which, all, are found among those of the series (2) and 
the product 

s = S'S" ■ ■ ■ S^ 

of which presents a substitution S with the help of which 
the form / will be transformed into a reduced form. 


Let us suppose now that two reduced forms / and /' 
be equivalent. 

If one of these forms, for example /, is interior to the 
domain R k , the form /' will also be interior to it. One 
concludes that the form / can be transformed into a form 
/' only with the help of a substitution which transforms 
the domain R k into itself. 

Let us suppose that the reduced equivalent forms / 
and /' be interior to the faces in p, dimensions of domains 
(!)• 


In this case one will declare supplementary condi¬ 
tions for the reduced forms / and /'. After having de¬ 
termined all the faces in p, dimensions of domains (1), 


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G. F. Voronoi (1908) 


one will choose a complete system of representatives of 
these various classes. Let us suppose that this system 
be formed by the faces in p dimensions 


Pi(ti), -P2<»> • • • > (3) 


Any positive quadratic form which is interior to a 
face in p dimensions of a domain of the set (J?) will be 
equivalent to a form which is interior to the faces (3), 
one will call it reduced. 

Two reduced positive quadratic forms which are in¬ 
terior to the faces (3) will be equivalent only provided 
that they be interior to the same face and that the sub¬ 
stitution which transforms one of them into another one 
also transforms this face into itself. 

We have arrived at the following result: 

A reduced quadratic form can be transformed into an¬ 
other reduced form or into itself only with the help of a 
substitution which transforms into itself a domain or a 
face of domains belonging to the series (1). 

Second Part 

Some applications 
of the general theory 
to the study of 
perfect quadratic forms 


On the principal perfect form 


We will not consider as different the quadratic forms 
of proportional coefficients, therefore one can arbitrarily 
choose the minimum value of a positive quadratic form. 

In that which follows, one will study only the perfect 
quadratic forms whose minimum is 1. One will indicate 
by D the determinant of these forms. 


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G. F. Voronoi (1908) 


Monograph Translations Series 


Among the various perfect forms, one form 


= x\ + x\ + • • • + X 2 n + X1X2 + X1X3 + . . . + X n -lX n 


t 


where 


1 


2 ’ 


da — 1 , ( i — 1 ; 2 , . . . , Ti ), 

(i = 1,2 ,n;j = 1,2, ... ,n\i j) 


and 


D 


n ■ 1 

2 n 


One will call principal the perfect form c p. 

The perfect form cp possesses n ( n + 1 l representations 
of the minimum 1, which define n ( n + 1 l linear forms 

Ai = £Ci, A2 = X2, ■ ■ ■ , 

\n — xji, A n +i — x 1 X21 A n _|_2 — «r 1 X3, . . ., 

A n(n+l) - X n —1 X n . 

2 


The domain R corresponding to the perfect form ip 
is made up of all the quadratic forms determined by the 
equality 


"(" + !) 

2 

^dijXiXj = ^2 Pk X l where p k > 0. 
k = 1 


(k= 1,2,.. 


n(n + 
2 ~ 


1 ) 


) 


From this equality one obtains 

Pk = aik + a 2k + • • • + a nk so long as k = 1, 2, . . ., n, 
p k = —dij so long as k > n; 

The form ip has been given for the first time by Zolot- 
areff in a Memoire titled: On an indeterminate equation 
of the third degree (in Russian) 

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G. F. Voronoi (1908) 

(i = 1, 2, . . ., n; j = 1, 2, . . ., n; i ^ j) 

therefore the domain R will be determined by the fol¬ 
lowing inequalities: 


dik + & 2 k + • • • + o-nk > 0, (k — 1 , 2, . . ., n) 

— a ij > 0. (* = 1,2, . . ., n ; j = 1,2,..., n \ i j) 


By virtue of (1), the perfect form ip possesses n ^ n + 1 l 
contiguous perfect forms which are determined by the 
equalities 

' Vfc = + PkX k (x 1 ,x 2 , ■ ■ ■ ,x n ), (k = 1,2, ... ,n) 

< ip k = ip - pkXiXj, ^fc = n + l,n + 2 ,..., n ( n + , (2) 

(* = 1,2 , . . . ,n-,j = 1,2, . . . , ra; i £ j) 


Let us find equivalent forms among the perfect forms 
contiguous to the perfect form ip. 

To this effect, let us determine the group g of sub¬ 
stitutions which do not change the form ip. 

Let us examine, in the first place, the form adjoint 
to the form ip. 

One will easily demonstrate that the coefficients of 
the form adjoint to ip are proportional to those of the 
form 

u; = + A 2 + • • • + A ^n+i) • (3) 


One concludes, by virtue of the theorem of Number 
17, that the principal perfect form ip is extreme. 

The quadratic form uj will have for expression 
uj = nx\ + nxl + • • • + n x^ — 2x±X2 — 2x1X3 — ... — 2x n -ix n 
where 

da — n, (i — 1 , 2, . . ., n j oq j — 1 - 


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G. F. Voronoi (1908) 


Monograph Translations Series 


(i = 1 , 2, . . . ,ra; j = 1 , 2, . . ., n; i ^ j) 


Let us find all the representations of the minimum 
of the form u>. 

The linear forms 

xi, x 2 , ■ ■ ■ , x n , x-i + x 2 + • • • + x n (4) 

characterise n + 1 representations of the value n of the 
form u>. 

I say that the form u has the minimum n and all 
the representations of the minimum of the form u> are 
characterised by the linear form (4). 

To demonstrate this, let us examine any one value 
cv(xi, X 2 , ■ ■ ■ , x n ) of the form u. By supposing that none 
of the numbers x±, X 2 , ■ ■ ■ , x n becomes zero, one will have 
by virtue of (13) 


u)(x 1 : x 2 , • • • , x n ) > n, 

the system x± = 1, X 2 = 1, ... ,x n = 1 being excluded. 

Let us suppose that any one of the numbers x±, X 2 , • • • , 
x n does not cancel out and that 

■Efc+i = 0) *Efc +2 = 0, . . ., x n = 0; 

one obtains, by virtue of (3), 

oo(x i, x 2 , . . ., x k , 0, . . . , 0) = 

(n - k + l)(xf + x% + . . . + xl) + ^2(x k - x h ) 2 , 
and it follows that 

u>(xi,x 2 ,. . . ,x n ) > k(n - k + 1), 


therefore 


oo(xi,X 2 ,...,x n )>n so long as k > 2. 


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G. F. Voronoi (1908) 


This stated, let us indicate by G the group of sub¬ 
stitutions which transform into itself the domain R. By 
virtue of (3), any substitution of the group G does not 
change the form oj . 

The group g being adjoint to the group G, one con¬ 
cludes that each substitution of the group g will only 
permute te linear forms (4) by changing the sign of a 
few among them. 

By noticing that 

x\ + x\ + . . . + x 2 n + (®1 + x 2 + ■ ■ ■ + X n ) 2 = 2 ip, 

one concludes that the group g is composed of all the 
substitutions which permute the forms 

x\ + x\ + . . . + x 2 n + (xi + x 2 + ■ ■ ■ + X n ) 2 . 


Let us indicate 


xq = —x\ — X 2 — ■ ■ ■ — x n and x' 0 = — x\ — x' 2 — ■ ■ ■ — x' n , (5) 

and let ko,ki,...,k n be any one permutation of numbers 
0, 1, 2, . . ., n. 

By posing 

Xi = eix' k . where e, = ±1, (i = 0, 1,2, . . ., n) (6) 
one will have 

Xo + X 1 + . . . + x n = e 0 x' ko + eix' kl + . . . + e n x' kn , 
and as, because of (5), 

Xo = Xi + . . . + x n = 0 and x' 0 , x[, . . . , x' n = 0, 
it is necessary that 


eo — ^ 1 — • • • — 

therefore the equalities (6) reduce to the one here: 

Xi = ex' k .. (i = 0, 1, 2, . . ., n; e = ±1) (7) 

70 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


The number of substitutions defined by the formulae 
obtained is equal to 2-1-2 ■■■ (n +1). By not considering as 
different two substitutions of opposite coefficients, one 
will say that the group g is composed of (ni)! different 
substitutions. § 

With the help of substitutions (7), one can transform 
any perfect form (2) contiguous to the principal form 
into another form contiguous to the form c p, arbitrarily 
chosen. 

We have arrived at the following important result. 

All the perfect forms contiguous to the principal per¬ 
fect form are equivalent. 


Let us choose one form among those of the series (2). 
Let us declare 


(pi = ip - px !X 2 - 


All the perfect form contiguous to the form ip are 
equivalent to the form <p\. 

Let us find the corresponding value of the parameter 
P- 


As we have seen in Number 22, the value searched 
for of p presents the smallest value of the function 


Vjxi, X 2 , ■ ■ ■ , X n ) - 1 
XiX 2 


( 8 ) 


determined with condition 


xix 2 > 0 . 


One will distinguish in the subsequent studies two 
cases: 

1). n = 2 and 2). n> 3. 

§ See: Minkowski, Zur Theorie der positiven quadratis- 
chen Formen [On the theory of the positive quadratic 
forms], (This Journal, V. 101, p. 200) 


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G. F. Voronoi (1908) 


First case: 

By comparing two n = 2 binary forms 

tp = xf + + X 1 X 2 and ip 1 = Xi X 2 X 1 X 2 — px 1 X 2 , 

one will notice that by making p = 2 one obtains the 
form 

<Pl = x\ + x\ - X1X2 

which is evidently equivalent to the perfect form ip, 
therefore the perfect form ip 1 is that which one has 
searched for. 

Second case: 

By making 

Xx = 1 , X 2 = 1 , X 3 = - 1 , 034 = 0 , = 0 , 

one obtains a value of the function (8) which is equal to 

1 . 


By making p= 1, one will present the form cpi under 
the following form: 

Fi = ^ [Oi, x 2 , ■ ■ ■ , X n ) 2 + (xx - X2) 2 + xj + . . . + x 2 n \ . (9) 


It results in that the form p>x is positive. On the 
ground of that which has been said in Number 23, one 
will find now all the systems of integers verifying the 
inequality 

ipx(xx,x 2 , • • •, X n ) < 1. 


By noticing that the inequality 

if l(2:i, X 2 , • • • , X n ) < 1 

is impossible, because the positive form ip 1 has integer 
values which corresponds to the integer values of vari¬ 
ables, one concludes that the form p>x is perfect. 


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G. F. Voronoi (1908) 


Monograph Translations Series 


With the help of the equality (9), one will easily de¬ 
termine all the presentations of the minimum of the per¬ 
fect form ip i. 


On the binary and ternary perfect forms and on the 
domains which correspond to them. 


The binary principal perfect form 

3 

p = x 2 + xy + y 2 , D = — 

possesses, as we have seen in Number 29, three contigu¬ 
ous perfect forms which are equivalent to the principal 
form. 

One concludes that all the perfect binary forms con¬ 
stitute only a single class of forms equivalent to the prin¬ 
cipal form. 

The domain 1Z corresponding to the principal form is 
made up of binary forms (a, b, c) which are determined 
by the equality 

ax 2 + 2 bxy + cy 2 = px 2 + p'y 2 + p"(x - y) 2 

where 

p>0, p' > 0, p” > 0 


It follows that the domain R is determined by the 
inequalities 

p = a + b > 0, p' = — b > 0, p" = c + b > 0. 


By calling reduced the positive binary forms verify¬ 
ing these inequalities, as we have done in Number 27, one 
will establish a well known method of reduction, due to 
Mr. Selling, f 

f Selling. Uber die binaren und ternaren quadratis- 
chen Formen. [On the binary and ternary quadratic 
forms] (This Journal, V. 77, p. 143) 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 73 



Monograph Translations Series G. F. Voronoi (1908) 

It results in that the domain R° is determined by the 
inequalities 

p = c — a > 0, p' = a + 2b > 0, p" = — b > 0. 


The inequalities obtained only differ from famous 
conditions of reduction of positive binary quadratic fo¬ 
rms due to Lagrange by the choice of the sign of the 
coefficient b, that which one can arbitrarily make in the 
method of Lagrange. | 


Let us examine now the ternary perfect forms. 

The principal perfect form 

¥> = x 2 + y 2 + z 2 + yz + zx + xy, D = \ 

possesses six contiguous perfect forms which, all, are 
equivalent to the perfect form 

<Pi = x 2 + y 2 + 2 2 + yz + zx 

which we have found in Number 31. 

The substitution 

x = —x', y = y 1 , z = -y' - z' 

transforms the form y>i into principal form. 

One concludes that all the ternary perfect forms form 
only a single class. 

J See: Lagrange. Recherches d’Arithmetique. [Studies 
in arithmetic] (Oeuvres de Lagrange published by Serret, 
V. Ill, p. 698) 

Gauss. Disquisitiones arithmeticae, art. 171. (Werke, 
V. I.) 

Lejeune Dirichlet. Vorlesungen fiber Zahlentheorie [Lec¬ 
tures on number theory], published by Dedekind, (Brau¬ 
nschweig 1894, §64, p. 155) 


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G. F. Voronoi (1908) 


Monograph Translations Series 


The domain R corresponding to the principal form is 
made up of all the ternary quadratic forms 

(a a' a "\ 

V b b' b") 


which are determined by the equality 

ax 2 -j- a'y 2 -j- a" z 2 -j- 2 byz -j- 2h'zx + 2b" xy = 

Pi + P 2 V 2 + P 3 Z 2 + P4(y ~ z) 2 + p 5 (z - x) 2 -\- p 6 (x - y) 2 . 


The domain R is determined by the inequalities 
pi = a -\- b' -\- b" > 0, 

p 2 = a' b" -j- b > 0, 
p 3 = a " b -)- b' > 0, 

P4 = —b > 0, 

p 5 = ~b' > 0, 
p 6 = -b" > 0. 

By calling reduced the positive ternary quadratic fo¬ 
rms belonging to the domain R, one will establish a 
method of reduction due to Mr. Selling. 

The domain R can be partitioned into 24 equivalent 
parts which can be transformed one into another with 
the help of 24 substitution adjoined to those which do 
not change the principal form. 

One of these parts, the domain 1Z, will be composed 
of all the ternary quadratic forms determined by the 
equality 

ax 2 + a'y 2 + a"z 2 + 2 byz 2 b'zx + 2 b"xy = 

pix 2 + p 2 y 2 + P 3 Z 2 + p 4 (y - z ) 2 + p 5 ^t + p [6 ]W 

where 

* = * 2 + y 2 + * 2 (y-z) 2 + (z~x) 2 , 
w = x 2 + y 2 + z 2 {y - z) 2 + (z - x) 2 + (x - y) 2 . 

God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 75 



Monograph Translations Series G. F. Voronoi (1908) 

One will determine the domain 1Z with the help of 
inequalities 

pi = a + 26' + b" > 0, 

P2 = a' b -j- b' -j- b" 7 > 0, 
p 3 = a " + b + b' + b" >0, 
p4. — —b -j - b' ^ 0 , 

P5 = — b' b" > 0, 
p 6 = -fe" > o. 


The domain 1Z enjoys the following properties: 

1. Any positive ternary quadratic form is equivalent 
to at least one form belonging to the domain 1Z. 

2. Two ternary quadratic forms which are interior to 
the domain 1Z can not be equivalent. 

By effecting the transformation of the domain 1Z with 
the help of all the substitutions of integer coefficients 
and of determinant ±1, one will make up the set (7 Z) of 
domains. 

Each domain 1Z belonging to the set (7 Z) possesses six 
domain contiguous by faces in 5 dimensions. 

The domain 1Z will be transformed into contiguous 
domain with the help of the following substitutions: 



76 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 



Each substitution of this series transforms into it¬ 
self a corresponding face in 5 dimensions of the domain 
1Z and permutes two domains of the set (1Z) which are 
contiguous through this face. 

This results in a method for the search for the sub¬ 
stitution which transforms a given form into a form be¬ 
longing to the domain 1Z. This method is analogous to 
that which has been shown in Number 27. 

By calling reduced any positive ternary quadratic 
form belonging to the domain 1Z, one will establish a 
new method of reduction of positive ternary quadratic 
forms which can be considered as a generalisation of the 
method of reduction of Lagrange. 


On the perfect form x\ x\x\ + * 1 X 3 + X 1 X 4 + 

...-(- X n — lX n . 


Let us examine the perfect form 

<Pi = + x\ + . . . + x 2 n + xxx 3 + xix 4 + . . . + x n -ix n 

obtained in Number 31. One has admitted 

Qjii — 1 j (i — ^12 — 0 5 ^ • 

(z = 1 , 2 , ... ,n-,j = 1,2 ,...,n\ij= 3 ) 

It results in that 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 77 



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G. F. Voronoi (1908) 


By supposing that n > 4, one will have n 2 — n repre¬ 
sentations of the minimum of the form ip\ the number of 
which is greater than n ( n +D . 

These representations of the minimum of the form ip i 
will be characterised by the linear forms 

A i — x i, A 2 — x 2 , ■ ■ ■ , A n — x n , 

An + l — X± &3, • • • 5 

A w(w+1) , - X n —\ Xfly A Tt(Tl-t-l) - X± X2 3^35 * * * 5 

, , ( 1 ) 

A „(„ + 1 ) | n 3 = XI + X 2 - x n , v ' 

A rt(rt-fl) j ^ ^ — Xi -(- X2 3^3 X4, ■ ■ ■ ) 

. A-n, 2 —ro — “(“ X 2 X n — i 'C-n,* 


The domain f?i corresponding to the perfect form ip\ 
is made up of forms determined by the equality 

n 2 — n 

f(x 1, x 2 , ■ ■ ■ , X n ) = Pk^l where p k > 0. 
k = 1 

(k = 1,2, . . . ,[n 2 - n]) 


Let us find the linear inequalities which define the 
domain R\. 

The number of these inequalities is so large in deed 
for n = 4. 

One will overcome the difficulties which result by the 
help of a particular method. 


Let us find the group g i of substitutions which do 
not change the form ip\. 

To this effect, let us introduce in our studies a quad¬ 
ratic form u determined by the equality 

^ _ 2 (Ai + A2 + • • • + A^ 2 _ n ). 


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G. F. Voronoi (1908) 


Monograph Translations Series 


After the reductions, one obtains 

cc( xi, x 2 , ■ ■ ■ , x n ) =nx\ + nx\ + 4^3 + . . . 

+ 4x 2 n + 2(n - 2)xix 2 - 4xxx 3 - . . . 
- 4xix n - 4x 2 x 3 - ... - 4x 2 x n . 


One can give in the form oj 2 the following expression: 


u(xi, X 2 , . . . , X n ) =(X! - x 2 ) 2 + (xi + x 2 ) 2 + 

( X 1 + x 2 - 2x 3 ) 2 + • • • + (X! + x 2 - 2x n ) 2 . 


It is easy to demonstrate that the form added to the 
perfect form ipi has coefficients which are proportional 
to those of the form 00 . 

It follows that the perfect form gp 1 is extreme. 

Let us observe that the linear form 

Xl + X 2 + . . . + X n , X 1 - X 2 , X 3 , X 4 , ■ ■ ■ , X n 

characterise n minimum 4 representations of the form 00 . 
In the case n > 5, other representations of the minimum 
of the form u do not exist; in the case n = 4, one obtains 
12 representations of the minimum of the form c 0 . 

By noticing that 

1 = ^ [(^1 + X 2 + . . . + X n ) 2 + (Xx - X 2 ) 2 + x\ + . . . + X 2 ] , 

one can say that the group g±, in the case n > 5, is com¬ 
posed of all the permutations of the forms 

(xi + X 2 + . . . + Xn) 2 , ( Xi-X 2 ) 2 , xl,..., x 2 n . 


In the case n = 4, one will determine by this method 
only divisor of the group g±. 

By indicating 


Ml = Xx + X 2 + . . . + X n , 


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G. F. Voronoi (1908) 


u2 = Xi - Xz, 

U3 — • • • 5 ^n — > 

u) = x\ + x ' 2 + ... + x' n , 

u 2 = X 1 — X 2 , 

u ' 3 = x' 3 , . . . ,u' n = x' n 

let us declare 

Ui = eiUf.., (i = 1, 2,. . ., n) (2) 

where = ±1 (z = 1,2, . . ., n) and the indices Aq, k 2 , ■ ■ ■ , k n 
present any one permutation of numbers l,2,...,n. 

Each system of equalities (2) defines a substitution 
of the group g\. 

One concludes that the group g± is composed of - 
2 n ~ 1 n! different substitutions, in the case n > 5. 


Let us suppose that the domain R 1 be determined by 
the inequalities 

^2Pij )a i > °- (fc = 1 , 2 , . . . , ct) 


By indicating 

^k(x i,x 2 , ■ ■ ■ ,x n ) = J2pffx*Xj, = 1 , 2 , • • •, a) 

one will determine, as we have seen in Number 22, a 
perfect forms 


¥>[ k) = Vi + Pk^k (k = 1,2, . . . , a) (3) 

contiguous to the perfect form ipi. 

All the substitutions of the group g± will make only 
one permutation of forms 

'Ll, ^ 2 , ■ ■ ■ , (4) 

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G. F. Voronoi (1908) 


Monograph Translations Series 


Let us effect the transformation of forms (3) and (4) 
with the help of the substitution 

Xi,x 2 .,x„ = x\, X! - x 2 = x' 2 , x 3 = x' 3 , ... , x n = x' n . (5) 

The series (4) will be transformed into a series 

* 1 , * 2 , ■■■,*'«,■ 


Let us indicate by g a group of substitutions 
xi 5 (i — 1 ) 2 ,..., n ) 

where e = ±1 (i = 1 , 2 , . . ., n) and Aq, & 2 , ■ ■ ., k n present a 
permutation of numbers l,2,...,n. Each substitution of 
the group g makes only one permutation of forms ( 6 ), 
and to a similar substitution corresponds a substitution 
of the group g\. 

By indicating 

V k (x i, a= 2 , • • •, x' n ) = ^P^x'iXj, (fc = 1 , 2 ,. . ., a) 
one will determine with the help of inequalities 

J^P^aiCLjy 0, (fc=l,2,...,a) (7) 

a domain 1Z. 

The form ip\ will be transformed into a form 

2 \ X 1 + X 2 + ■ ■ ■ + X n ), 

with the help of the substitution (5), and any system 
(xi,X 2 , . . . , x n ) of integers xi,X 2 ,---,x n will be replaced 
by a system (x^, x' 2 , • • • , x ' n ) of number, also integer, x' x , x 2 , 

. . . , x' n satisfying the condition 

x[ + X 2 + . . . + x' n = 0(mod 2). ( 8 ) 

It results in that the linear forms (1) which corre¬ 
spond to the various representations of the minimum of 
the form <pi will be replaced by the forms 

x'i + x) and x\ - x) (i = 1 , 2 , . . . , n; j = 1 , 2 , . . ., n; i ^ j) 
God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 81 



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which characterise the various representations of the m- 
inimum 2 of the quadratic form x' 2 + x' 2 2 + . . . + x' n 2 , in 
the set ( X ') of all the systems (x ' x , x' 2 , ■ ■ ■, x' n ) of integers 
x \, x ' 2 , • ■ ■ , x' n satisfying the condition (8). 

One concludes that the edges of the domain 1Z will 
be characterised by the quadratic form 

(xl + x)) 2 and (x\ - x)) 2 . 

(i = 1,2 , ,n; j = 1,2 ,. . ., n; i j) 


By virtue of (7), one obtains the inequalities 

+ 2Pj k) + P {k) > 0 and P} z k) - 2 pj k } + pj k) > 0. 

(fe = l,2,...,CT;z=l,2,...,n;j = l, 2 ,...,n;Z 7 ^j) 

( 9 ) 

Let us examine any one form 

^'(x'i,x 2 , . . . , x' n ) = Pijx'iX j (10) 

belonging to the series (6). By virtue of (9), one will 
have 

Pa 2 Pi j Pjj > 0 and Pa — 2 Pij -I - Pjj ^ 0. 

(i = l,2,...,n;j = l,2,...,n;i^j) (11) 


Among these conditions one will find t quantities 
which define the coefficients of the form (10) to an im¬ 
mediate common factor. All these equalities will be of 
the form 


Pkk 2CkhPkh Phh — 0 where Ckh — il. (12) 


Let us suppose that there exists a combination of 
values of k and h satisfying the conditions 


Pkk + 2 Pkh + Phh > 0 and Pkk ~ 2Pkh + Phh > 0. (13) 

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G. F. Voronoi (1908) 


Monograph Translations Series 


By noticing that the coefficient Pkh does not enter 
the other inequalities (11), one concludes that the coef¬ 
ficient Pkh remains undetermined. 

For all the coefficients of the form (10) to be deter¬ 
mined by the conditions (12) to an immediate common 
factor, it is necessary, the coefficient Pkh being indepen¬ 
dent of other coefficients, that all the coefficients which 
remain cancel out. 

By virtue of inequalities (13), this supposition is im¬ 
possible, therefore the inequality (12) has to hold for all 
the values of indices k and h. 

One obtains n ^ n ~ 1 '> conditions 

P k k ~ 2 e k hPkh + Phh = 0 where e k h = ±1. 

(fe = l,2,...,n;h=l,2,...,n;fe^=h) (14) 

which serve to determine the coefficients P k h in functions 
of coefficients 

Piii P 22 , ■ ■ ■ , Pnn- (15) 

The coefficients P 11 , P 22 , ■ ■ ■ , Pnn can not be indepen¬ 
dent, and will be connected by at least n— 1 equations of 
the form (12). Therefore, in at least n— 1 case, one will 
have the equations of the form 

Pkk ± 2P k h + Phh = 0. (16) 


To make short we will call these equations double. 

This stated, let us suppose, in the first place, that 
there exists at least one coefficient among those of the 
series (15) which does not enter in the double equations 
(16). One can suppose, to fix the ideas, that Pn be such 
a coefficient. 

The coefficient Pn being independent, all the coeffi¬ 
cients P22, ■ ■ ■ , P nn will cancel each other and, by virtue 
of (14), the coefficients 

B23, -P24, • • • , P n -l,n 


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G. F. Voronoi (1908) 


will also cancel one another out. 

The coefficient P±i is used for determining the coef¬ 
ficients P 12 , P13, ■ ■ ■ , Pin with the help of equations (14) 
which take the form 

Pn — ZeikPik = 0; (k = 2, 3, . . ., n) 
it follows that 

2Pik = eifePn. (k = 2, 3n) 


As, on the ground of the supposition made, 
Pn + 2eikPik > 0, (fc = 2, 3, . . . , n) 


it is necessary that 


P li > 0, 


and one can declare 


P ii = 1. 


The form (10) is determined by the equalities ob¬ 
tained, and one will have 


(x I , X 2 , 


i) — x'i + ei2X l 1 x' 2 + • • • + ei r 


x[ x'n 


(17) 


By replacing the variables 

612^2, ^13X3, • • • , & i n X n 

by the variables x' 2 , ■ ■ ■ , x' n , one will replace the form (17) 
by the form 

(x h X 2 i • • • : X 7l ) — X1 (x 1, X 2 j • • • , X n ) . 


Let us suppose, in the second place, that all the co¬ 
efficients (15) enter in the double equations (16). 

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G. F. Voronoi (1908) 


Monograph Translations Series 


At least one of the coefficients (15) is not zero. Let 
us suppose that Pkk 0. Following the hypothesis, the 
coefficient Pkk enters in at least one double equation 

P k k±2P kh + P hh = 0. 


It follows that 


l’kh — 0 and Pkk Phh — 0) 

therefore the coefficients Pkk and P^h are of opposite 
signs. Let us suppose, to fix the ideas, that 

Pn = -1. (18) 


By examining the inequalities 

Pi i ± 2 Pi k Pkk > 0, (fc = 2,3,...,n) 
one deduces 

Pkk > 0. (k = 2,3, . . . ,n) 


It results in that the double equation 


Pkk ± 2 P k h + Phh = 0 


has to be impossible so long as k > 2 and h > 2, therefore 
all the double equations will be of the form 

Pii±2Pi k + P kk = 0. (fc = 2,3, ...,n) 


From these equations one gets, by virtue of (18), 

Pkk = 1 and Pik = 0 . (fc = 2, 3, . . ., n) (19) 


By substituting the values obtained of coefficients 
Ph,P 2 2,..., Pnn in the equations 


Pkk ~ 2e kh Pkh + Phh = 0 where e kh = ±1, 

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G. F. Voronoi (1908) 


one obtains, because of (19), 

F*kh —— &kh where e^kh —— il- 
(k = 2,3, ..., n; h = 2,3, ..., n; k h) 


The form (10) will have for expression 
^\ x 'i,x' 2 , . . . ,x' n ) = -x' 2 + x' 2 2 + x' 3 2 + . . . 

+ + 2e23x' 2 x' 3 + 2e24®2a;4 + . . . 

^ e n-l,n x n -l X n 


where 


e 2 3 = ±1) e 2 4 = ±1, . . ., e n -i }n = ±1. 


( 20 ) 


„ _ (n — l)(n — 2 ) , . „ _ 

One obtains in this way 2 2 different forms. By 

permuting the variables and by changing their signs, one 
will particularly decrease the number of various forms 
determined by the formula (20). 


With the help of results obtained, one can easily 
recognise whether a given quadratic form J2 a ij x i x j be¬ 
longs to the domain 1Z or not. 

One will examine, in the first place, the sums 

eifeOife + e 2 fca 2 fc + ...-)- e n kd n k 

where e lk = ±1, e 2fc = ±1, . . . , e nk = ±1 and e k k = 1- (k = 

1,2,..., n) 

All these sums have to be positive or zero. The in¬ 
equalities 


®kk |0Hfc| • • • |^fc — l,fc| 1 ,/c | • • • l^ra/cj ^ 0, 

(fc=l,2,...,n) (21) 

present the conditions necessary and sufficient for the 
inequalities 


eifcOtift -(- €-2k® J 2k ^-nk^nk ^ 0 

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G. F. Voronoi (1908) 


Monograph Translations Series 


(k = 1,2, . . . ,n) 

to hold. 

Let us examine, in the second place, the inequalities 

— a 11 + <*22 + 033 + • • • + a nn + 2e23 a 23 + 2e24®24 +••• 

+ 2e n _i jn a n _i ;n > 0 

where 

e 2 3 = ±1, e 2 4 = ±1, • • •, e n _i >n = ±1, 

These inequalities can be replaced by a single one 

“On + O 22 + O 33 + . . . + a nn — 2 | 023 | “ 2|a24| “ • • • “ 2 | <3. ^— 1,7T. | > 0- 


One will present this inequality under the form 
on + a 22 + • • • + a nn 2 |a 12 
> 2 (an 

By permuting the variables, one obtains n inequali¬ 
ties 

On + O 22 + • • • + o nn — 2 |ai 2 | — 2|ai3| — ... — 2|a n _i >n | 

> 2 (afcfc — | a i fc | — . . . — |a n fc|) where k = 1,2 , ,n 

(22) 

We have arrived at the following result. One can 
easily recognise whether a given positive quadratic form 
/ belong to the domain f?i or not. To this effect, one will 
transform the form / by a form /' with the help of the 
substitution adjoint to the substitution (5) and one will 
examine 2n inequalities (21) and (22). For the form / to 
belong to the domain R i, it is necessary and sufficient 
that the form /' verifies 2 n inequalities (21) and (22). 

38 

Let us return now to the perfect forms (3) contiguous 
to the perfect form tp±. We have seen that these forms 


• • • 2|a n _i jn 
■ la. 12 1 - ... - lain I) • 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 87 



Monograph Translations Series G. F. Voronoi (1908) 

will be transformed with the help of the substitution (5) 
into forms 

\ [x'x 2 + x ' 2 2 + ... + x' n 2 ) + Pk^' k {x i, ai'a, • • • ,x' n ). 

(k= 1,2,..., a) 


The forms T' 2 , . . . , T' CT can be transformed with the 
help of substitutions belonging to the group g into forms 


' !)• x'zi-x'x - x' 2 + x' 3 + x' 4 + . . . + x' n ), 

- x' 2 2 + x\ 2 + x' 3 + . . . + x' n 2 - 2 x' 1 x' 3 - . . . 

. 2XiX n “1“ 2634 X 3 X 4 “j" • ■ • 2 e n -i jn X n _ 1 X n , 


(23) 


where 


C 34 Tlj ■ ■ ■ ■ €-71 — 1,71 Tl. 


The inverse substitution to the substitution (5): 
x'l = Xi + X 2 + • • • + x n , X 2 — X1 X 2 , X3 x 3 ,, x' n = x n 

will transform the forms (23) into forms 

1 ) . - 2 xix 3 

2 ) . 4 (xix 2 - S 3 4X 3 x 4 - ... - ix n ), 

where; 634 = 0 or 1 , ... , 6 n -i t7l = 0 or 1 . 


One concludes that all the perfect forms contiguous 
to the form cpi are equivalent to the following perfect 
forms 


1) -¥>i - PX 1 X 3 , 

2) .</?i d" p(xix 2 634X3X4 ... 6 n —i, n x n — 1 x n ), 


where 


634 = 0 or 1, . . . , 6 n - i, n = 0 or 1. 


Study of the perfect form y>i — px 1X3. 


88 


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G. F. Voronoi (1908) 


Monograph Translations Series 


The perfect form ip i, possesses, as we have seen in 
Number 38, many contiguous perfect forms which are 
not equivalent. 

One will determine in the following only a single per¬ 
fect form 

¥>2 = <Pi ~ px±x 3 

contiguous to the perfect form yq. 

We have demonstrated in Number 22 that the pa¬ 
rameter p presents the smallest value of the function 

Oi, x 2 , ■ ■ ■ , x n ) - 1 

P = Fi -——- ( 1 ) 

X i X 3 

determined on condition that 

x±x 3 > 0. (2) 


By declaring 

33 1 = 1, X 2 = 0, a3 3 = l, 33 4 = -l, 335 = 0,..., X n = 0, 

one obtains the value of the function (1) which is equal 
to 1, therefore 

0<p<l. (3) 

Let us effect the transformation of the function (1) 
with the help of the substitution 

33 3 = x 'l 1 


— 33 1 + 33 2 = x‘ 2 , 

Xl + 33 2 + • • • + X n = -33'g, 
X4 = X4 , . . . , x n = x n 

one will have 

= 33' 1 2 + 33' 2 2 + .. , + 33' n 2 -2 

P -X’ x ( 33 [ + X' 2 + . . . + X’ n ) 

where, because of (2), 

Xi(x '1 + x' 2 + . . . + x ' n ) <0 


(4) 

(5) 

( 6 ) 


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Monograph Translations Series 


G. F. Voronoi (1908) 


and, because of (4), 

x\ + x ' 2 + . . . + x' n = O(mod 2), 
the variables x\ + x ' 2 + . . . + x' n being integers. 

Let us indicate 

f(x 1 ,x 2 ,---iX n )=xl+xl + ...+Xn+px 1 (x 1 + X2 + ... + x n ). 


By virtue of (5) and (6) the value looked for of p is 
defined by the conditions that the inequality 

f(xi,x 2 , • • • , x n ) < 2 

is impossible, so long as the integers x±, x 2 , ■ ■ ■ , x n verify 
the congruence 


Xi + X 2 + . . . + x n = O(mod 2), (7) 

and that there exists at least one system (Zi, l 2 , . . ., l n ) 
verifying the equation 

f(xi,x 2 , • • •, x n ) = 2 (8) 

and the congruence (7). 

The form / can be determined by the equality 


( gp 1 \ 2 / 

+ {X 3 + P — J + 
+ [x n + p^-j +^l + p- 


n — 1 


P x ± . 


( 9 ) 


It follows that the form / will be positive, provided 


that 


1 + P 


n — 1 


-p 2 > o, 


and the upper limit R of values of p verifies the equation 

n — 1 


1 + R 


- R 


1, 


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G. F. Voronoi (1908) 
therefore 


Monograph Translations Series 


R = 2 
\fn - 1. 


( 10 ) 


This presented, let us examine a system (l i, l 2 , . . . , l n ) 
of integers verifying the equation (8) and the congruence 
(7). 


I say that there will be the inequalities 


; jl h 
il+p 2 


< 1. (i = 2, . . . ,n) 


( 11 ) 


In effect, if one suppose that 

< 1, 


Ik + P~2 


one will determine e k = ±1 such that the inequality 


Ik + 2 ek + — 


< 


Ik + P~2 


holds, and one will present 

l'i = li and l' k = l k = 2efc. (i = 1 , 2 , . . . , n; i k) 


The condition (7) will be satisfied, and one will have, 
by virtue of (9), the inequality 

/(z'i,r 2 ,...,o <2, 

which is contrary to the hypothesis. 

By examining the inequalities (11) and the form / 
with the help of the formula (9), one will easily demon¬ 
strate that among the system of integers verifying the 
equation (8) with condition (7) is found at least one sys¬ 
tem (Zi, 1 2 , . . . , l n ) satisfying the conditions 

f(h,h,...,ln) = 2 (12) 

God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 91 



Monograph Translations Series 


G. F. Voronoi (1908) 


and 

l 2 = l 3 + 5, I 3 = I 4 = . . . = l n where (5 = 0 or ±1. (13) 

By virtue of ( 6 ), one will have the inequality 
£1 [^i + <5 + (n — 2 ) 13 ] < 0 . 

One can suppose that 

h<0, (14) 

and it follows that 

£i + <5+(ti — 2)^3 >0, 

therefore, because of (13) and (14), it is necessary that 

h > 0 . 

I say that 1 3 = 1. To demonstrate this, let us ef¬ 
fect the transformation of the positive quadratic form 
f (xi, X 2 , ■ ■ ■, x n ) with the help of the substitution 

xi = -x, x 2 = y, x 3 = x 4 = • • • = x n - z-, (15) 

one will obtain a ternary positive form 

F(x, y, z) = x 2 + y 2 + (n - 2)z 2 - px(-x + y + (n - 2)z). 

By virtue of the condition (7), the integers x,y,z ver¬ 
ify the congruence 

x + y + (n — 2 )z = 0 (mod 2 ). (16) 

By indicating 

u = —l 1 , v = 1 2 , w = l 3 , 

one will have, because of (12), (13) and (15), 

F(u, v,w) = 2, 

92 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) Monograph Translations Series 

and the condition (16) will be fulfilled. 


The inequality 

f( x ,y,z ) < 2 

is impossible so long as the integers x,y,z verify the con¬ 
gruence (16). 

Let us effect the transformation of the form F(x,y,z ) 
with the help of the substitution 

x = x, + y'+ (n - 2)z', y = x' - y 1 , z = z 1 . (17) 


The set of systems ( x,y,z ) of integers verifying the 
congruence (16) will be replaced by the set of systems 
( x',y',z ') of arbitrary integers. 

Let us indicate by F 1 (x', y ', z 1 ) the transformed form. 
Let D and D' be the determinants of forms F(x,y,z ) and 
F'(x', y', z'). By virtue of (17), one will have 

D' = 4D. (18) 


Let us notice that the number 2 presents the mini¬ 
mum of the form obtained F'(x 1 , y ', z 1 ) determined in the 
set of all the systems (x 1 , y' , z ') of integers, the system 
(0,0,0) being excluded. 

On the ground of the known theorem § on the limit of 
the minimum of a ternary positive quadratic form, one 
will have the inequality 

2 < yfl2D 7 . 

§ See: Gauss. Werke, V. II, p. 192, Gottingen 1863. 

Lejeune-Dirichlet. Uber die Reduktion der positiven 
quadratischen Formen mit drei unbestimmten ganzen 
Zahlen. (This Journal, V. 40, p. 209) 

Hermite. Sur la theorie des formes quadratiques ter- 
naires. [On the theory of ternary quadratic forms] (This 
Journal, V. 40, p. 173) 


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Monograph Translations Series 


G. F. Voronoi (1908) 


It follows that 

D' > 4, 

and because of (18), one obtains 

D >1. (19) 


This presented, let us observe that the form F(x, y, z ) 
has the following values: 


F(u,v,w) = 2, F( 1,1,0) = 2, F(l,-l,0) = 2 + 2p. 


By transforming the form F(x, y, z) with the help of 
the substitution 


(u, 1 , 1 \ 

v, 1, -1 , (20) 

\w, 0, 0 / 

one obtains a form 


F 0 (x', y', z') 


ax 


J 2 


+ a'y' 2 + a"z rz + 2 by'z' + 2 b'z'x' + 2b"x'y', 


^ // _/2 


where 

o = 2, a 1 = 2, a" = 2 + 2 p and b = p. (21) 


The product a a' a" in any positive ternary quadratic 

form ^ ^ is, as one knows, always greater than 

the determinant of the form, unless the coefficients b, b ', 
b" do not simultaneously cancel one another out. 

By indicating with Do the determinant of the form 
Fq(x', y' , z 1 ), one will have, because of (21), 

D 0 < 4(2 + 2 p), 

and as, by virtue of (20), 


D 0 = 4 w 2 D, 


it becomes 

w 2 D < 2 + 2 p. 


94 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


By virtue of (3) and (19), one obtains the inequality 

w 2 < 4, 

therefore 

w = 1. 

By returning to the equalities (13), one obtains 
1 1 = — u, I2 = S and I3 = 1, I4 = 1, . . . , l n = 1, 


where 


u> 0 and 5 = 0,1,2. 


By substituting the values found of Z 1 , Z 2 , 
function (5), one will have 


P = 


u 2 + S 2 + n — 4 
u( — u + 5 + n — 2) 


, l n in the 


( 22 ) 


It remains to determine the smallest value of this 
function providing that 

u > 0, — u S n — 2>0, u = n + 5(mod2) and 5 = 0,1,2. 

(23) 


Let us admit 

u = \fn — 1 + a, 
a being a real number. 

The function (22) takes the form 

_ 2n + (2a —2) v /n + a 2 —2a + 5 2 —3 

P ij/n + (a — 2)n + (5 — 2a)y / n + 1 — a 2 + a5 — 5 


(24) 


ity 


The value searched for of p has to verify the inequal- 

P < R, 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 95 



Monograph Translations Series 


G. F. Voronoi (1908) 


therefore because of (10), one will have 


2 


- 1 


- p > 0. 


(25) 


After the reductions, one obtains 

2 

s/n- 1 ~ P ~ 

(1 — <5 2 — (- 2 <5 — a 2 )\fn 5 2 — 25 — 1 — a 2 — 2a 2a5 
(\fn — 1) [n\fn + (a — 2)n + (5 — 2 a)\/n + 1— a 2 + a5 — (5] 

and, because of (25), it becomes 

(1 - 5 2 + 25 - a 2 )^/n + 5 2 - 25 - 1 - a 2 - 2a + 2a5 > 0. 


By noticing that 

5 2 - 25 - 1 - a 2 - 2a + 2a5 <0 so long as <5 = 0, 1,2, 
one obtains the inequality 

1 - 5 2 + 25 - a 2 >0. 


By making (5 = 0 and 2, one will have 

a 2 < 1 as long as (5 = 0 and 2 (26) 


By making 5=1, one will have 

a 2 < 2 so long as 5=1. (27) 


Let us indicate by m a positive integer determined 
with the help of inequalities 

\fn — 1 < m < \fn. (28) 


By declaring 

n = m 2 + p, (29) 

96 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 




G. F. Voronoi (1908) 


Monograph Translations Series 


one will have a positive integer p verifying the inequali¬ 
ties 

0 < p < 2m + 1. (30) 

First case: p is an odd number. 

By virtue of (23) and (29), one will have a congru¬ 
ence 

uErn 2 -|-p + i5(mod2), 
p being an odd number; one can declare 

u = m 2 + 5 + 1 + 2t. (31) 


By declaring 


Vn = m + £, 


one will have 

0<€<1, (32) 

because of (28). By virtue of (24), one obtains the equal¬ 
ity 

u = m — 1 + £ + a, 
and because of (31), it becomes 

£ cx =■ m ^ — m —[— 2 —(— 2t -(- 5. (33) 


By supposing that (5 = 0 or 2, one obtains 
£ + a = 0(mod 2). 

By virtue of (26) and (32), it is necessary that 

4 + a = 0, 

therefore 

u = m — 1 so long as (5 = 0 and 8 = 2. 

By supposing that 5 = 1, one obtains, because of 
(33), 

£ + a = l(mod 2). 

God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 97 



Monograph Translations Series G. F. Voronoi (1908) 

By virtue of (27) and (32), the integer £ + a can have 
only two values 

£ + a = ±1, 

and it results in that 

u = m or m — 2 as long as 5=1. 


One obtains four values of the function (22): 

(to — l) 2 + ra — 4 _ TO 2 + n —3 


Pi 

P3 


(m—l)(n —m—1) ’ 
(m — 2) 2 -|-n — 3 


P2 

P4 = 


m(n — m — 1 ) ’ 
(m — 2) 2 + n 


(m—2) (n —m-f-1) 5 ( m — l)(n — ra + 1) 

among which is found the smallest value looked for of p. 


By noticing that 

_ _ Pj2_ 3_ 

P 1 P 2 m(m — l)(n — m — 1) ’ 

_ _ 2p+ 2 _ 

^ 1 (m — l)(n — m — l)(n — m + 1) ’ 

_ P + 1 _ 

(m — l)(m — 2)(n — m + 1) ’ 

one obtains, because of (30), 

Pi < P4 < P3 

and 

p 2 < Pi so long as p > 3, 

Pi < P 2 so long as p < 3. 


There exists only a single odd value of p verifying 
the inequalities 0 < p < 3, therefore one will have the 
inequality 

Pi < p 2 so long as p = 1. 


We have arrived at the following result. 


The smallest value of p will have for expression 


m 2 + n — 3 
m(n — m — 1), 


(34) 


98 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) Monograph Translations Series 

provided that n = m 2 + p, and the odd number p verifies 
the inequalities 

3 < p < 2m + 1. 

In the case n = m 2 + 1, the smallest value of p will be 

_ (m — l) 2 + n — 4 
P (m — 1) (n — m — 1) 

Second case: p is an even number. 

One will have, because of (23), the inequality [sic] 
u = m 2 + <5(mod 2). 


By presenting 

u = m 2 + (5 + 2 1, 
one will have the equalities 

u = m — l+£ + a and £ + a = m 2 — m -\- 2t f- 8 1. 


By supposing that (5 = 0 or 2, one obtains 


and it follows that 


€ + a — 11 


u = m so long as <5 = 0 and 2. 


By supposing that (5=1, one obtains 
£ + a = 0 or £ + a = 2, 


therefore 

u = m 1 or u = m -\- 1 so long as (5=1. 


The smallest value of p is found among the following 
values of the function (22): 


_ m 2 + n — 4 

hi m{n — m — 2) ’ 

_ (m —l) 2 -t~n —3 

P3 (m — !)(«. — m) ’ 


P2 


P4 


fm + l) 2 + 7(. —3 
(m + l)(n —m —2) 1 

_ m 2 + n 

m(n — m ) 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 99 



Monograph Translations Series 


G. F. Voronoi (1908) 


By noticing that 


P 2 - Pi 
Pi ~ p4 
P4 ~ Pz 


2m + 4 — p 

m(m + 1) (n — m — 2) ’ 
4m — 2p 

m(n — m)(n — m — 2)’ 
2m — p 

m(m — 1) (n — m) ’ 


one obtains, because of (30), 


Pz < P4 < Pi < p 2 • 


We have arrived at the following result: 

The smallest value of p is expressed by the equality 

(m— l) 2 + n — 3 

P = ---- 

(m — 1) (n — m) 

provided that n = m 2 + p, and the even number p verifies 
the inequalities 

0 < p < 2m + 1. 


We have determined the value of the parameter p 
which defines the perfect form <pi + px 1X3. The determi¬ 
nant D of this form, by virtue of (4) and (9), will have 
for expression 


D 


4 -)- 4p — (n — 1 )p 2 


(35) 


The corresponding value of the function M(aij) de¬ 
fined in Number 16 will be 


M(aij ) 


Y 4 + 4p - (n - l)p 2 ' 

By applying the formulae obtained to the case: 


n = 4, 5, 6, 7, 8, 

100 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) Monograph Translations Series 

one obtains the same value of p 


P = 1 - 


The corresponding perfect forms will be 


[ 1 —I | lO [r-1 | ^ [r-H | CO [tH | CN 

lO c© 


CN CN CN CN 



3 





3 

1-1 

II 


«l« q 


II II II II «Z 

Q Q Q Q I 

C*3 



to 

CO 

1> 

+ 

8 


H 

H 

CO 


ID 

CD 


8 

H 


H 


+ 

+ 

+ 

+ 

+ 










8 

+ 

+ 

+ 

+ 

8 


■S' 



4- 

8 

8 

8 

8 

i 

oq oo 

8 

8 

8 

8 

8 

+ 

+ 

+ 

+ 


(M 

oq ID 

Oq CD 

oq c~ 


8 

8 

8 

8 

+ 





<N <M 





H 

+ 

+ 

+ 

+ 

+ 

<N <M 

oq oq 

oq <m 

oq oq 

<N tH 


8 

8 

8 

H 

+ 

+ 

+ 

+ 


<M iH 

Oq TH 

oq th 

oq th 



8 

8 

8 



One comes across all these perfect forms in the Mem- 
oire of Mr.’s Korkine and Zolotareff: Sur les formes 

quadratiques. [On the quadratic forms] J 


£ Mathematische Annalen, V. VI, p. 367. 

God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 101 



Monograph Translations Series 


G. F. Voronoi (1908) 


The formulae obtained give a mean for the study of 
various perfect forms which verify the inequality 

M(aij) > 2 . 


By making, for example, n = 12, one will have 
m = 3 and p = 3. 


By virtue of (34), one obtains 

3 

P= 4’ 

therefore, because of (35), 



1 

212 > 


and it follows that 


12 [16 

= 2 y— > 2. 


All the extreme forms studied by Mr.’s Korkine and 
Zolotareff do not give a function M(aij) of values which 
exceed 2. 


On the quadratic perfect forms and on the domains 
which correspond to them. 


We have seen in Number 29 that to the quaternary 
principal perfect form 

if = x l + xl + xl + xl -\- XiX 2 -\- XiX 3 -\- XiX^f- X 2 X 3 -\- X 2 X4-\- X3X4, 

r> — 5 

corresponds the domain R made up of forms 
pixj + p 2 xl + p 3 x% + p 4 xl + P5(xi - x 2 ) 2 + P[ 6] (xi - a3 3 ) 2 + 

102 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) Monograph Translations Series 

P 7 (X 1 -X4) 2 + P8(X2-X 3 ) 2 + P9(X2-X4) 2 + P10(X 3 -X4) 2 ■ 


All the perfect forms contiguous to the principal form 
<p are equivalent to the form 

<Pi = x 1 + x\ + x\ + x\ X±X 3 X1X4 X2X 3 X2X4 -)- X 3 X4, 

D = b 

The corresponding domain f?i is made up of forms 

Pix\ + P2xj + p 3 X% + P4X4 + p 5 (Xi - X 3 ) 2 + p 6 (x 1 - X 4 ) 2 + 

pr(x 2 - x 3 ) 2 + p 8 (x 2 - X4) 2 + p 9 ( x 3 - x 4 ) 2 + 
Plo(xi + X 2 - x 3 ) 2 + Pn(xi + X2 - X 4 ) 2 + 
p 12 ( 2:1 + X 2 - x 3 - X4) 2 . 


Let us examine the perfect forms contiguous to the 
perfect form ip\. 

We have demonstrated in Number 38 that all these 
forms are equivalent to the forms 

1) . - pxix 3 , 

2) - + p(x\X 2 — 5x 3 X4), where (5 = 0 or 1. 


Let us examine three perfect forms 
!)• 'Pi + px\X 2 , 2). ipx-px 1 X 3 , 3). pi + p(xix 2 - x 3 x 4 )- 


1) . By making p = 1 in the form + px 1 X 2 , one 
obtains the principal perfect form ip. 

2) . Let us notice that the form p\— px\x 3 is equivalent 
to the form p>\ + px\X 2 - 

In effect, the substitution 

X\ = —x'x, X2 = x' 3 , X 3 = x' 2 , X4 = x'x + x'4 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 103 



Monograph Translations Series G. F. Voronoi (1908) 

does not change the form tp± and transforms the form 
X 1 X 2 into the form —x^x^. 

3). By making p = 1 in the form tp± + p(x 1 X 2 — X3X4 ), 
one obtains the form 

x\ + x\, 33§ + x\ + X\X2 + X1X3 + 331 X 4 + X 2 X 3 + X 2 X 4 
which is evidently equivalent to the perfect form yq. 

One concludes that all the perfect forms contiguous 
to the perfect form <pi are equivalent to the forms and 
Wi¬ 


lt follows that the set of all the quaternary perfect 
forms be divided into two classes represented by the per¬ 
fect forms <p and <p ±. 

The set (f?) of domains corresponding to various qua¬ 
ternary perfect forms is made up of two classes, too, 
represented by the domains R and f?i. 


On the perfect forms in five variables and on the do¬ 
mains which correspond to them. 

We have determined two perfect forms in five vari¬ 
ables 

if = x\ + xl + . . . + 33 § + 331332 + 33 i 33 3 + . . . + X 4 X 5 , D = ^, 
Wl = X 1 + x 2 + ■ • ■ + x 5 + X 1 X 3 + X 1 X 4 + • • • + 334335 , D = 

The corresponding domains R and R± will be com¬ 
posed of forms 

R) Pix\ + p 2 xl + . . . + p 5 xl + p 6 (x 1 - 33 2 ) 2 + 

P7(X 1 - X 3 ) 2 + . . . + P 15 (£4 - 335 ) 2 , 

Rl) Pix\ + p 2 xl + . . . + p 5 xl + p 6 (xi - X 3) 2 + . . . 

+ P2o(a3l +332-334- X 5 ) 2 . 


104 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


Examine the perfect forms contiguous to the perfect 
face ipi. We have demonstrated in Number 38 that all 
these forms are equivalent to the forms 

1 ) . <pi-px 1 x 3 , ^ 

2) . ipi — p(x±X2 — 6x3X4 — S' X3X5 — 5" X4X5). 

where 


(5 = 0 or 1 , 8' = 0 or 1 , 5" = 0 or 1 . 


In the second case one obtains 8 perfect forms. By 
permuting the variables x 3 , X4, X5 one will replace the 
forms (1) by 4 forms; thus all the perfect forms con¬ 
tiguous to the perfect form <pi are equivalent to the 5 
following forms: 

1. pi + px iX 2 , 

2 . ipi + px1X3, 

3. ipi + p(xix 2 - X 4 X 5 ), 

4 ■ ipi + p(xix 2 - X3X5 - X4X5) , 

5. ipi + p(x 1 x 2 - X3X4 - x 3 x 5 - x 4 x 5 ). 


1) . By making p = 1 in the perfect form ip 1 + px±x 2 , 
one obtains the perfect form p. 

2) . We have seen in Number 42 that the perfect form 
Pi — px\x 3 is determined by the value p = 1 of the param¬ 
eter p in the case n = 5 . One obtains the form 

Pi = x\ + x\ + x\ + x\ + x\ + X 1 X 4 + xix 5 + • • • + 2 : 42:5 (2) 

which will be transformed with the help of the substitu¬ 
tion 


XI = —X2, x 2 = x\ — x' 2 , X3 = x' 3 , X4 = X 2 + x' 4 , X5 = x' 2 + x' 5 
into a perfect form pi. 

God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 105 



Monograph Translations Series G. F. Voronoi (1908) 

3 ) . In the form <pi + p(x 1X2 — X4X5 ), one will put p = 0 
and one will obtain the form 

x\ + x\ + . . . + X% + X1X2 + X\X 3 + . . . + X 3 X 5 
which is evidently equivalent to the form <pi. 

4 ) . In the form <pi + p(xix 3 — x 3 x 3 — X4X5) ), one will put 
p = 1 and one will obtain the form 

x\ + x\ + . . . + x 2 5 + XxX2 + XxX 3 + . . . + X2X5 
which is evident to the perfect form (2) 

5 ) . It remains only to determine the perfect form: 

T>1 + p(xix 2 - X 3 x 4 - X 3 x 5 - X 4 x 5 ). ( 3 ) 

By effecting the transformation with the help of the 
substitution 

-x 1 + x 2 = x' 1: 

Xl + x 2 + x 3 + X4 + x 5 = x' 2 , 

X 3 = x' 3 , X4 = x' 4 , x 5 = x' 5 ( 4 ) 

of the form 

2<pi + 2p(xix 2 - x 3 x 4 - x 3 x 5 - x 4 x 5 ), 
one obtains the form 

/ 2 1 / 2 1 / 2 1 / 2 1 / 2 I 

Xl + X 2 + x 3 + x 4 + x 5 + 

2 + ^2 + +2:4 +*5 - 2 a 3 2 CC 3 - 2X2X4- 

2 x' 2 x' 5 — 2 x' 3 x' 4 — 2 x' 3 x' 5 — 2 x' 4 x' 5 ] . ( 5 ) 

By virtue of ( 4 ) the integer variables x' 4 , x' 2 , x' 3 , x' 4 , x' 5 
verify the congruence 

x'i + x' 2 + x' 3 + x' 4 + x' 5 = 0(mod2). (6) 

106 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


By applying to the form (5) the method unveiled in 
Number 2, one will determine the value of the upper 
limit R > 0 of value § with the help of equations 

Ci — f?€i7 = 0, 

C2 + 2 - C3 - £4 - C5) = 0 , 

£3 + R (—£2 + C3 - £4 - Cs) = 0, 

£4 + f?(—£ 2 — C3 + £4 — Cs) = 0, 

C5 + R(—£ 2 — C3 — C4 + Cs) = 0 . 


It results in that 

C2 = C3 = £4 = £5, 
and one obtains the equations 

Ci (1 — R) = 0 and ^ 2(1 — 21?) = 0, 

thus 



By declaring 

cci = 0, x' 2 = 1, x' 3 = 1, x ' 4 = 1, ai'g = 1, (7) 

one will satisfy the condition ( 6 ) and one will have the 
value 4 — 4p of the form (5). 

By making 
one obtains 


4 — 4p = 2, 


P= 2‘ 


It follows that the positive quadratic form 

/ 2 . / 2 . . / 2 . 

+ £2 + • • • + X 5 + 

4 [-£Ci + a ; 2 + a ; 3 + £4 + x 5 - 2 ir 2 ir 3 - . . . - 2 a 3 4 a ; 5 
will have a value 2 corresponding to the system (7). 
God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 


( 8 ) 


107 



Monograph Translations Series 


G. F. Voronoi (1908) 


By virtue of that which has been discussed in Num¬ 
ber 23, the smallest value of the form ( 8 ) will correspond 
to a system (Zi, Z 2 , ■ ■ ■ , Is) verifying the inequality 

l\ + ^2 + I 3 + I 4 + ^5 < 2 • —- y where R = —. 

-LX, ^ ^ 


One obtains the inequality 


i? + ii + il + /2 + z§<4. 


It is easy to demonstrate that the system (7) is the 
only one verifying this inequality on condition ( 6 ), the 
systems which verify the inequality 

-aq + aq + 3:3 + 3:4 + a : 5 - 2x 2 x 3 - 

2x 2 x' 4: — 2x 2 x' 5 — 2x' 3 x' 4: — 2 x' 3 x' 5 — 2x' 4 x' 5 > 0 

being excluded. By making p = \ in the form (3), one 
obtains the perfect form 


<Pz = a:i + x 2 + . . . + x ' 5 + -aqaq + aqa : 3 + . . . 


-\-x2X5 + ^x 3 x 4 + ^x 3 x 5 + ^x 4 x 5 , 


D 


1 

25 ' 


The corresponding domain R 2 is composed of forms 


Pix\ + p 2 a:i + • • • + P 5 X 5 + Pe(x! - x 3 ) 2 + . . . + pn (x 2 - x 5 ) 2 
+P 12 ( 3:1 + a: 2 - a; 3 - x 4 ) 2 + p 13 ( 2:1 + a; 2 - a; 3 - a; 5 ) 2 
+p 14 ( 3:1 + x 2 - x 4 ~ X5) 2 + pi 5 ( —aq - a: 2 + a; 3 + 3:4 + a; 5 ) 2 . 


The number of parameter pi, P 2 , ■ ■ ■ , P 15 being equal 
to the number of dimensions of the domain R 2 , one will 
determine without trouble 15 inequalities which define 
the domain f? 2 . 


108 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


We have demonstrated that all the perfect forms con¬ 
tiguous to the perfect form ip 4 are equivalent to the per¬ 
fect forms ip,<pi and ip 2 . 

Choose the perfect forms contiguous to the perfect 
form ip 2 . 

To this effect let us notice, in the first place, that 
the perfect form ip\ is contiguous to the perfect form ip 2 , 
then observe that all the perfect forms contiguous to the 
form ip2 are equivalent. 

To demonstrate this, examine all the faces in 14 di¬ 
mensions of the domain R 2 . 

The domain R 2 is characterised by 15 quadratic form- 

s 

rp 2 ™ 2 2 ~ 2 

1 j **'2 > ^ 3 ’ ^ 4 ’ 

(xi — X3) 2 , (xi — x 4 ) 2 , (xi - X5) 2 , 

< (x 2 - X3) 2 , (x 2 - x\), (x 2 - X 5 ) 2 , (9) 

(xi + X 2 - X 3 - X4) 2 , (xi + x 2 - x 3 - x 5 ) 2 , 

, (xi + X 2 - X 4 - X5) 2 , (-X! - X 2 + X 3 + X 4 + X 5 ) 2 - 


Each face in 14 domains of the domain R 2 possesses 
14 of these, and the form which remains can be called 
form opposite to the face. 

One concludes that each face is well determined by 
the opposite face. 

For the perfect forms contiguous to the perfect form 
ip 2 to be equivalent, it is necessary and sufficient that all 
the faces of the domain R 2 could be transformed one to 
one with the help of substitutions which do not change 
the domain R 2 . 

It would be easy to write all these substitutions, but 
one will proceed in another way, more speedy. 

Let us observe that the face P belonging to the do¬ 
main R 1 and R 2 is characterised by all the forms (9), the 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 109 



Monograph Translations Series 


G. F. Voronoi (1908) 


form (— x\ -X2 + X3 + X4 + X5) 2 being excluded. 

With the aid of substitution associated with the sub¬ 
stitution ( 4 ), one will replace the forms ( 9 ) by the forms: 

' (*; ± x ' 2 ) 2 , (*; ±*') 2 , (*i ±x' 4 ) 2 , ±x' 5 y, 

< (*' + x') 2 , (x' 2 + x' 4 ) 2 , (x' 2 + x' ) 2 , (x' 3 + x^) 2 , (x' 3 + X[ 5 ]') 2 , 

. (A + x' 5 ) 2 , (X 2 + x' 3 + x' 4 + x'g) 2 . 

(10) 

By changing the sign of x'i and by permuting the vari¬ 
ables x' 2 , x 3 , x' 4 , x' 5 , one will transform into itself the 
forms (10), and the form (x' 2 + x 3 + x 4 + A) 2 wi 11 not 
change. 

To each similar substitution corresponds a substitu¬ 
tion which transforms into itself the domain R 2 and the 
face P of the domain R 2 , and does not change the form 
(-X1 - x 2 + X 3 + x 4 + X 5 ) 2 . 

By changing the sign of x’ x and by permuting x 2 , x 3 , 
x 4 , x' 5 , one will transform the form (x' x + x 2 ) 2 into forms 

(x'i ± x') 2 , (x' ± x') 2 , (x' ± x' 4 ) 2 , (x' ± x') 2 
and one will transform the form (x 2 + x 3 ) 2 into forms 

(x' 2 + x') 2 ,(x'+x' 4 ) 2 ,(x'+x') 2 , 

(x' 3 + x 4 ) 2 , (x' 3 + x' [5] ) 2 , (x 4 + A) 2 . 

Thus only the forms 

(x'i + x') 2 , (x' 2 + x') 2 , (x' 2 + x' + x' 4 + A) 2 (11) 

remain to examine. 

By returning to the forms ( 9 ), one obtains the forms 
corresponding to the forms (11). 

x 2, A, (-X1 - x 2 + X 3 + x 4 + x 5 ) 2 . ( 12 ) 

It is demonstrated that all the forms ( 9 ) can be trans¬ 
formed into forms (12) with the help of substitutions 
which do not change the domain R 2 . 

110 7 th February, 2007 God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 



G. F. Voronoi (1908) 


Monograph Translations Series 


With the help of substitutions 

Xi = x ' 2 - x' 5 , 

X 2 = x' 3 , 
x 3 = x' 2 , 

X 4 = x\ + x ' 2 — x ' 4 — x' 5 , 

X 5 = —x\ + x ' 3 

and 

xi = x\, 

x 2 = -x\ - x 2 + x ' 3 + x ' 4 + x' 5 , 
x 3 = x 3 , 

X 4 — X 4 , 
x 5 = x' 5 , 

one will transform the domain R 2 into itself, and the 
form x 2 will be transformed into forms x 3 and (— x ' 4 — x' 2 -\- 
x' 3 + A + x ' 5 ) 2 . 

We have demonstrated that all the forms of the do¬ 
main R 2 are equivalent. It results in, from that we have 
seen, that all the perfect forms contiguous to the perfect 
form ip 2 are equivalent to the perfect form ip 

One concludes that all the perfect forms in five vari¬ 
ables constitute three different classes represented by 
the perfect forms ip, ip i and ip 2 . 

The set of domains (f?) can be divided into three 
classes also, represented by the domains R, R\, and R 2 . 

End of the first Memoire. 


God’s Ayudhya’s Defence, Kit Tyabandha, Ph.D. 7 th February, 2007 111